ࡱ> G02 !"#$%&'()*+,-./ACI q  A C NWYbdfsuwyLNPXZ\^` b!!N""'` bjbj 4%2nnn8Rnv,2td.~vK"(d-4cccccccfh|hc9aaac.dYYYacYacYYHM"~ @"nLlN$Dd0tdL $i$iM$iM,aaYaaaaaccXaaatdaaaa222d222222 5.1 Extrema and the Mean Value Theorem Learning Objectives A student will be able to: Solve problems that involve extrema. Study Rolles Theorem. Use the Mean Value Theorem to solve problems. Introduction In this lesson we will discuss a second application of derivatives, as a means to study extreme (maximum and minimum) values of functions. We will learn how the maximum and minimum values of functions relate to derivatives. Lets start our discussion with some formal working definitions of the maximum and minimum values of a function. Here is an example of a function that has a maximum at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20a" \* MERGEFORMATINET and a minimum at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20d" \* MERGEFORMATINET : Definition A function  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET has a maximum at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20a" \* MERGEFORMATINET if  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28a%29%20%5Cge%20f%28x%29" \* MERGEFORMATINET for all  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x" \* MERGEFORMATINET in the domain of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f." \* MERGEFORMATINET Similarly,  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET has a minimum at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20a" \* MERGEFORMATINET if  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28a%29%20%5Cle%20f%28x%29" \* MERGEFORMATINET for all  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x" \* MERGEFORMATINET in the domain of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f." \* MERGEFORMATINET The values of the function for these  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x-" \* MERGEFORMATINET values are called extreme values or extrema.  INCLUDEPICTURE "http://www.ck12.org/ck12/images?id=123722" \* MERGEFORMATINET  Lets recall the Min-Max Theorem that we discussed in lesson on Continuity. Observe the graph at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20b" \* MERGEFORMATINET . While we do not have a minimum at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20b" \* MERGEFORMATINET , we note that  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28b%29%20%5Cle%20f%28x%29" \* MERGEFORMATINET for all  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x" \* MERGEFORMATINET near  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=b." \* MERGEFORMATINET We say that the function has a local minimum at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20b." \* MERGEFORMATINET Similarly, we say that the function has a local maximum at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20c" \* MERGEFORMATINET since  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28c%29%20%5Cge%20f%28x%29" \* MERGEFORMATINET for some  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x" \* MERGEFORMATINET contained in open intervals of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=c." \* MERGEFORMATINET  Min-Max Theorem: If a function  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29" \* MERGEFORMATINET is continuous in a closed interval  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=I%2C" \* MERGEFORMATINET then  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29" \* MERGEFORMATINET has both a maximum value and a minimum value in  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=I." \* MERGEFORMATINET In order to understand the proof for the Min-Max Theorem conceptually, attempt to draw a function on a closed interval (including the endpoints) so that no point is at the highest part of the graph. No matter how the function is sketched, there will be at least one point that is highest. We can now relate extreme values to derivatives in the following Theorem by the French mathematician Fermat. Theorem: If  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28c%29" \* MERGEFORMATINET is an extreme value of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET for some open interval of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=c%2C" \* MERGEFORMATINET and if  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%28c%29" \* MERGEFORMATINET exists, then  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%28c%29%20%3D%200." \* MERGEFORMATINET  Proof: The theorem states that if we have a local max or local min, and if  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%28c%29" \* MERGEFORMATINET exists, then we must have  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%28c%29%20%3D%200." \* MERGEFORMATINET  Suppose that  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET has a local max at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20c." \* MERGEFORMATINET Then we have  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28c%29%20%5Cge%20f%28x%29" \* MERGEFORMATINET for some open interval  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%28c%20-%20h%2C%20c%20%2B%20h%29" \* MERGEFORMATINET with  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=h%20%3E%200." \* MERGEFORMATINET  So  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20f%28c%2Bh%29-f%28c%29%5Cle%200." \* MERGEFORMATINET  Consider  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Clim_%7Bh%20%5Cto%200%5E%2B%7D%20%5Cfrac%7Bf%28c%2Bh%29-f%28c%29%7D%7Bh%7D" \* MERGEFORMATINET . Since  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20f%28c%2Bh%29-f%28c%29%5Cle%200" \* MERGEFORMATINET , we have  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Clim_%7Bh%5Cto%200%5E%2B%7D%20%5Cfrac%7Bf%28c%2Bh%29-f%28c%29%7D%7Bh%7D%5Cle%5Clim_%7Bh%5Cto%200%5E%2B%7D%200%3D0." \* MERGEFORMATINET  Since  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%28c%29" \* MERGEFORMATINET exists, we have  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20f%27%28c%29%3D%5Clim_%7Bh%5Cto%200%7D%5Cfrac%7Bf%28c%2Bh%29-f%28c%29%7D%7Bh%7D%3D%5Clim_%7Bh%5Cto%200%5E%2B%7D%20%5Cfrac%7Bf%28c%2Bh%29-f%28c%29%7D%7Bh%7D" \* MERGEFORMATINET , and so  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%28c%29%5Cle%200." \* MERGEFORMATINET  If we take the left-hand limit, we get  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20f%27%28c%29%3D%5Clim_%7Bh%5Cto%200%7D%20%5Cfrac%7Bf%28c%2Bh%29-f%28c%29%7D%7Bh%7D%3D%5Clim_%7Bh%5Cto%200%5E-%7D%20%5Cfrac%7Bf%28c%2Bh%29-f%28c%29%7D%7Bh%7D%5Cge%200." \* MERGEFORMATINET  Hence  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%28c%29%20%5Cge%200" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%28c%29%20%5Cle%200" \* MERGEFORMATINET it must be that  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%28c%29%20%3D%200." \* MERGEFORMATINET  If  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20c" \* MERGEFORMATINET is a local minimum, the same argument follows. We can now state the Extreme Value Theorem. Definition We will call  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20c" \* MERGEFORMATINET a critical value in  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Ba%2Cb%5D" \* MERGEFORMATINET if  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%28c%29%20%3D%200" \* MERGEFORMATINET or  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%28c%29" \* MERGEFORMATINET does not exist, or if  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20c" \* MERGEFORMATINET is an endpoint of the interval. Extreme Value Theorem: If a function  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29" \* MERGEFORMATINET is continuous in a closed interval  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Ba%2Cb%5D" \* MERGEFORMATINET , with the maximum of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20c_1" \* MERGEFORMATINET and the minimum of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20c_2%2C" \* MERGEFORMATINET then  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=c_1" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=c_2" \* MERGEFORMATINET are critical values of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f." \* MERGEFORMATINET  Proof: The proof follows from Fermats theorem and is left as an exercise for the student. Example 1: Lets observe that the converse of the last theorem is not necessarily true: If we consider  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%20x%5E3" \* MERGEFORMATINET and its graph, then we see that while  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%280%29%20%3D%200" \* MERGEFORMATINET at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%200%2C" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%200" \* MERGEFORMATINET is not an extreme point of the function.  INCLUDEPICTURE "http://www.ck12.org/ck12/images?id=123705" \* MERGEFORMATINET  Rolles Theorem: If  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET is continuous and differentiable on a closed interval  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Ba%2Cb%5D" \* MERGEFORMATINET and if  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28a%29%20%3D%20f%28b%29%2C" \* MERGEFORMATINET then  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET has at least one value  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=c" \* MERGEFORMATINET in the open interval  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%28a%2Cb%29" \* MERGEFORMATINET such that  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%28c%29%3D0%20" \* MERGEFORMATINET . The proof of Rolle's Theorem can be found at  HYPERLINK "http://en.wikipedia.org/wiki/Rolle's_theorem"  HYPERLINK "http://en.wikipedia.org/wiki/Rolle's_theorem" http://en.wikipedia.org/wiki/Rolle's_theorem. Mean Value Theorem: If  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET is a continuous function on a closed interval  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Ba%2Cb%5D" \* MERGEFORMATINET and if  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27" \* MERGEFORMATINET contains the open interval  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%28a%2Cb%29" \* MERGEFORMATINET in its domain, then there exists a number  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=c" \* MERGEFORMATINET in the interval  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%28a%2Cb%29" \* MERGEFORMATINET such that  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28b%29-f%28a%29%20%3D%20%28b%20-%20a%29%20f%27%28c%29." \* MERGEFORMATINET  Proof: Consider the graph of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET and secant line  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=s" \* MERGEFORMATINET  as indicated in the figure.  INCLUDEPICTURE "http://www.ck12.org/ck12/images?id=123767" \* MERGEFORMATINET  By the Point-Slope form of line  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=s" \* MERGEFORMATINET  we have  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=y%20-%20f%28a%29%20%3D%20m%28x%20-%20a%29" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=y%20%3D%20m%28x%20-%20a%29%20%2B%20f%28a%29." \* MERGEFORMATINET  For each  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x" \* MERGEFORMATINET in the interval  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%28a%2Cb%29%2C" \* MERGEFORMATINET let  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=g%28x%29" \* MERGEFORMATINET be the vertical distance from line  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=S" \* MERGEFORMATINET to the graph of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f." \* MERGEFORMATINET Then we have  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=g%28x%29%20%3D%20f%28x%29%20-%20%5Bm%28x%20-%20a%29%20%2B%20f%28a%29%5D" \* MERGEFORMATINET for every  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x" \* MERGEFORMATINET in  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%28a%2Cb%29." \* MERGEFORMATINET  Note that  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=g%28a%29%20%3D%20g%28b%29%20%3D%200." \* MERGEFORMATINET Since  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=g" \* MERGEFORMATINET is continuous in  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Ba%2Cb%5D" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=g%27" \* MERGEFORMATINET exists in  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%28a%2Cb%29%2C" \* MERGEFORMATINET then Rolles Theorem applies. Hence there exists  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=c" \* MERGEFORMATINET in  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%28a%2Cb%29" \* MERGEFORMATINET with  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=g%27%28c%29%20%3D%200." \* MERGEFORMATINET  So  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=g%27%28x%29%20%3D%20%20f%27%28x%29%20-%20m" \* MERGEFORMATINET for every  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x" \* MERGEFORMATINET in  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%28a%2Cb%29." \* MERGEFORMATINET  In particular,  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=g%27%28c%29%20%3D%20f%27%28c%29%20-%20m%20%3D%200" \* MERGEFORMATINET  and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=f%27%28c%29%20%26%20%3D%20m%5C%5C%0Af%27%28c%29%20%26%20%3D%20%5Cfrac%7Bf%28b%29-f%28a%29%7D%20%7Bb-a%7D%5C%5C%0Af%20%28b%29-f%28a%29%20%26%20%3D%20%28b-a%29%20f%27%28c%29." \* MERGEFORMATINET  The proof is complete. Example 2: Verify that the Mean Value Theorem applies for the function  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%20x%5E3%20%2B%203x%5E2%20-%2024x" \* MERGEFORMATINET on the interval  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5B1%2C4%5D." \* MERGEFORMATINET  Solution: We need to find  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=c" \* MERGEFORMATINET in the interval  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%281%2C4%29" \* MERGEFORMATINET such that  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%284%29%20-%20f%281%29%20%3D%20%284-1%29%20f%27%28c%29." \* MERGEFORMATINET  Note that  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%28x%29%20%3D%203x%5E2%20%2B%206x%20-%2024%2C" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%284%29%20%3D%2016%2C" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%281%29%20%3D%20-20." \* MERGEFORMATINET Hence, we must solve the following equation:  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=36%20%26%20%3D%203f%27%28c%29%5C%5C%0A12%20%26%20%3D%20f%27%28c%29." \* MERGEFORMATINET  By substitution, we have  Since we need to have  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=c" \* MERGEFORMATINET  in the interval  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%281%2C4%29%2C" \* MERGEFORMATINET the positive root is the solution,  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20c%3D%5Cfrac%7B-2%20%2B%20%5Csqrt%7B52%7D%7D%20%7B2%7D%5Capprox%202.61" \* MERGEFORMATINET . Lesson Summary We learned to solve problems that involve extrema. We learned about Rolles Theorem. We used the Mean Value Theorem to solve problems. Multimedia Links For a video presentation of Rolle's Theorem (8.0), see  HYPERLINK "http://www.youtube.com/watch?v=2_FCYzch8ww" Math Video Tutorials by James Sousa, Rolle's Theorem (7:54) INCLUDEPICTURE "http://i3.ytimg.com/vi/2_FCYzch8ww/hqdefault.jpg" \* MERGEFORMATINET . For more information about the Mean Value Theorem (8.0), see  HYPERLINK "http://www.youtube.com/watch?v=RHQvYvL679o" Math Video Tutorials by James Sousa, Mean Value Theorem (9:52) INCLUDEPICTURE "http://i3.ytimg.com/vi/RHQvYvL679o/hqdefault.jpg" \* MERGEFORMATINET . For an introduction to LHpitals Rule (8.0), see  HYPERLINK "http://www.youtube.com/watch?v=PdSzruR5OeE" Khan Academy, LHpitals Rule (8:51) INCLUDEPICTURE "http://i1.ytimg.com/vi/PdSzruR5OeE/hqdefault.jpg" \* MERGEFORMATINET . For a well-done, but unorthodox, student presentation of the Extreme Value Theorem and Related Rates (3.0)(12.0), see  HYPERLINK "http://www.youtube.com/watch?v=9YIKOLlnRYo" Extreme Value Theorem (10:00) INCLUDEPICTURE "http://i2.ytimg.com/vi/9YIKOLlnRYo/hqdefault.jpg" \* MERGEFORMATINET . Review Questions In problems #13, find the coordinates of all absolute and relative extrema. (Units on the axes indicate 1 unit). Continuous on  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5B0%2C9%5D" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/images?id=123766" \* MERGEFORMATINET  Continuous on  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5B0%2C9%5D" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/images?id=123760" \* MERGEFORMATINET  Continuous on [0, 4) U (4, 9)  INCLUDEPICTURE "http://www.ck12.org/ck12/images?id=123750" \* MERGEFORMATINET  In problems #46, find the absolute extrema on the closed interval indicated, and sketch the graph.  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%20-%20x%5E2%20-%206x%20%2B%204%2C" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5B-4%2C1%5D" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%20x%5E3%20-%20x%5E4%2C" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5B0%2C2%5D" \* MERGEFORMATINET   EMBED Equation.DSMT4 ,  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5B-2%2C0%5D" \* MERGEFORMATINET  Verify Rolles Theorem for  EMBED Equation.DSMT4  by finding values of x for which  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 . Verify Rolles Theorem for  EMBED Equation.DSMT4 . Verify that the Mean Value Theorem works for  EMBED Equation.DSMT4  on the interval [1, 2]. Prove that the equation  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%5E3%20%2B%20a_1%20x%5E2%20%2B%20a_2%20x%20%3D%200" \* MERGEFORMATINET  has a positive root at x = r, and that the equation  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=3x%5E2%20%2B%202a_1%20x%20%2B%20a_2%20%3D%200" \* MERGEFORMATINET  has a positive root less than r. Review Answers Absolute maximum (7, 7); absolute minimum (4, -1); relative maximum (2, 5) and (7, 7); relative minimum (0,4), (4, -1), and (9, 3).  INCLUDEPICTURE "http://www.ck12.org/ck12/images?id=123745" \* MERGEFORMATINET  Absolute maximum (7, 9); absolute minimum (9, 0); relative maximum (0, 6) and (7, 9); relative minimum (3,1.5) and (9, 0).  INCLUDEPICTURE "http://www.ck12.org/ck12/images?id=123726" \* MERGEFORMATINET  Absolute minimum (0, 1); relative minimum (0, 1); there is no max since the function is not continuous on a closed interval.  INCLUDEPICTURE "http://www.ck12.org/ck12/images?id=123721" \* MERGEFORMATINET  Absolute maximum (-3, 13); absolute minimum (1, -3)  INCLUDEPICTURE "http://www.ck12.org/ck12/images?id=123747" \* MERGEFORMATINET  Absolute maximum  EMBED Equation.DSMT4 , absolute minimum (2, -8)  INCLUDEPICTURE "http://www.ck12.org/ck12/images?id=123765" \* MERGEFORMATINET  Absolute minimum at  EMBED Equation.DSMT4   INCLUDEPICTURE "http://www.ck12.org/ck12/images?id=123741" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20f%28x%29%3D0" \* MERGEFORMATINET  at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20x%3D0%2C%5Cpm%202" \* MERGEFORMATINET .  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20f%27%28x%29%3D0%2C%20x%3D%5Cpm%20%5Cfrac%7B2%5Csqrt3%7D%7B3%7D%3B" \* MERGEFORMATINET  by Rolles Theorem, there is a critical value in each of the intervals (-2, 0) and (0, 2), and we found those to be  EMBED Equation.DSMT4 .  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%202" \* MERGEFORMATINET  at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20-1%2C0" \* MERGEFORMATINET .  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%28x%29%20%3D%200" \* MERGEFORMATINET  at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%5Capprox%20-0.84" \* MERGEFORMATINET ; by Rolles Theorem, there is a critical value in the interval (-1, 0) and we found it to be  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%5Capprox%20-0.48." \* MERGEFORMATINET  Need to find  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=c%20%5Cin%20%281%2C2%29" \* MERGEFORMATINET  such that  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20f%282%29-f%281%29%3D%282-1%29f%27%28c%29%3Bc%3D%5Csqrt%202." \* MERGEFORMATINET  Let  EMBED Equation.DSMT4 . Observe that  EMBED Equation.DSMT4 . By Rolles Theorem, there must exist  EMBED Equation.DSMT4  such that  EMBED Equation.DSMT4 . Rolles and MVT Practice Determine whether Rolles Theorem can be applied to EMBED Equation.DSMT4 on the interval EMBED Equation.DSMT4 . If Rolles Theorem can be applied, find all values of  EMBED Equation.DSMT4 in the interval  EMBED Equation.DSMT4  such that EMBED Equation.DSMT4 . 1.  EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4  3.  EMBED Equation.DSMT4  4.  EMBED Equation.DSMT4  5.  EMBED Equation.DSMT4  6.  EMBED Equation.DSMT4  7.  EMBED Equation.DSMT4  Determine whether the Mean Value Theorem can be applied to EMBED Equation.DSMT4 on the interval  EMBED Equation.DSMT4 . If MVT can be applied, find all values of EMBED Equation.DSMT4 in  EMBED Equation.DSMT4  such that  EMBED Equation.DSMT4 . 8.  EMBED Equation.DSMT4  9.  EMBED Equation.DSMT4  10.  EMBED Equation.DSMT4  11.  EMBED Equation.DSMT4  12.  EMBED Equation.DSMT4  Getting at the Concept: 13. Let  EMBED Equation.DSMT4  be continuous on  EMBED Equation.DSMT4  and differentiable on EMBED Equation.DSMT4 . If there exists  EMBED Equation.DSMT4  in  EMBED Equation.DSMT4  such that  EMBED Equation.DSMT4 , does it follow that  EMBED Equation.DSMT4  Explain. 14. When an object is removed from a furnace and placed in an environment with a constant temperature of 90F, its core temperature is 1500F. Five hours later the core temperature is 390F. Explain why there must exist a time in the interval when the temperature is decreasing at a rate of 222F per hour. Answers: 1.  EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4  3. Rolles Theorem cannot be applied to  EMBED Equation.DSMT4  since  EMBED Equation.DSMT4  is not differentiable at  EMBED Equation.DSMT4  which is on  EMBED Equation.DSMT4  4.  EMBED Equation.DSMT4  5.  EMBED Equation.DSMT4  6. Rolles Theorem cannot be applied since  EMBED Equation.DSMT4  is not continuous at  EMBED Equation.DSMT4  which is on  EMBED Equation.DSMT4 . 7. Rolles Thrm cannot be applied since  EMBED Equation.DSMT4  8.  EMBED Equation.DSMT4  9.  EMBED Equation.DSMT4  10.  EMBED Equation.DSMT4  11.  EMBED Equation.DSMT4  12. MVT cannot be applied since  EMBED Equation.DSMT4  is not continuous @  EMBED Equation.DSMT4  which is on  EMBED Equation.DSMT4  13. No. Ex:  EMBED Equation.DSMT4  14. MVT,  EMBED Equation.DSMT4  Rolles Theorem & Mean Value Theorem HW Determine if Rolles Theorem can be applied to  EMBED Equation.DSMT4  on [a, b]. If it can, then find all values of  EMBED Equation.DSMT4  such that  EMBED Equation.DSMT4 . 1.)  EMBED Equation.DSMT4  2.)  EMBED Equation.DSMT4  3.)  EMBED Equation.DSMT4  4.)  EMBED Equation.DSMT4  Determine if the MVT can be applied to  EMBED Equation.DSMT4  on [a, b]. If it can, then find all values of  EMBED Equation.DSMT4  such that  EMBED Equation.DSMT4 . 5.)  EMBED Equation.DSMT4  6.)  EMBED Equation.DSMT4  7.)  EMBED Equation.DSMT4  Answers! 1.) x = 1 2.)  EMBED Equation.DSMT4  3.) Cannot apply Rolles Thm. (not diff.) 4.)  EMBED Equation.DSMT4  5.)  EMBED Equation.DSMT4  6.)  EMBED Equation.DSMT4  7.)  EMBED Equation.DSMT4  5.2 The First Derivative Test Learning Objectives A student will be able to: Find intervals where a function is increasing and decreasing. Apply the First Derivative Test to find extrema and sketch graphs. Introduction In this lesson we will discuss increasing and decreasing properties of functions, and introduce a method with which to study these phenomena, the First Derivative Test. This method will enable us to identify precisely the intervals where a function is either increasing or decreasing, and also help us to sketch the graph. Note on notation: The symbol  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cepsilon" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cin%20" \* MERGEFORMATINET are equivalent and denote that a particular element is contained within a particular set. We saw several examples in the Lesson on Extreme and the Mean Value Theorem of functions that had these properties. If  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x_1%29%20%3C%20f%28x_2%29" \* MERGEFORMATINET whenever  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x_1%20%3C%20x_2" \* MERGEFORMATINET for all  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x_1%2C%20x_2%20%5Cin%20%5Ba%2Cb%5D%2C" \* MERGEFORMATINET then we say that  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET is strictly increasing on  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Ba%2Cb%5D." \* MERGEFORMATINET If  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x_1%29%20%3E%20f%28x_2%29" \* MERGEFORMATINET whenever  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x_1%20%3E%20x_2" \* MERGEFORMATINET for all  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x_1%2C%20x_2%20%5Cin%20%5Ba%2Cb%5D%2C" \* MERGEFORMATINET then we say that  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET is strictly decreasing on  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Ba%2Cb%5D." \* MERGEFORMATINET  Definition A function  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET is said to be increasing on  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Ba%2Cb%5D" \* MERGEFORMATINET contained in the domain of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET if  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x_1%29%20%5Cle%20f%28x_2%29" \* MERGEFORMATINET whenever  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x_%7B1%7D%20%5Cle%20x_%7B2%7D" \* MERGEFORMATINET for all  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x_1%2C%20x_2%20%5Cin%20%5Ba%2Cb%5D." \* MERGEFORMATINET A function  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET is said to be decreasing on  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Ba%2Cb%5D" \* MERGEFORMATINET contained in the domain of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET if  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x_1%29%20%5Cge%20f%28x_2%29" \* MERGEFORMATINET whenever  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x_1%20%5Cge%20x_2" \* MERGEFORMATINET for all  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x_1%2C%20x_2%20%5Cin%20%5Ba%2Cb%5D." \* MERGEFORMATINET  Example 1: The function  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%20x%5E3" \* MERGEFORMATINET is strictly increasing on  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%28-%5Cinfty%2C%20%2B%5Cinfty%29" \* MERGEFORMATINET :  INCLUDEPICTURE "http://www.ck12.org/ck12/images?id=123705" \* MERGEFORMATINET  Example 2: The function indicated here is strictly increasing on  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%280%2Ca%29" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%28b%2Cc%29%2C" \* MERGEFORMATINET and strictly decreasing on  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%28a%2Cb%29" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%28c%2Cd%29." \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/images?id=123722" \* MERGEFORMATINET  We can now state the theorems that relate derivatives of functions to the increasing/decreasing properties of functions. Theorem: If  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET is continuous on interval  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Ba%2Cb%5D%2C" \* MERGEFORMATINET then: If  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%28x%29%20%3E%200" \* MERGEFORMATINET for every  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%5Cin%20%5Ba%2Cb%5D%2C" \* MERGEFORMATINET then  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET is strictly increasing in  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Ba%2Cb%5D." \* MERGEFORMATINET  If  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%28x%29%20%3C%200" \* MERGEFORMATINET for every  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%5Cin%20%5Ba%2Cb%5D%2C" \* MERGEFORMATINET then  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET is strictly decreasing in  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Ba%2Cb%5D." \* MERGEFORMATINET  Proof: We will prove the first statement. A similar method can be used to prove the second statement and is left as an exercise to the student. Consider  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x_1%2C%20x_2%20%5Cin%20%5Ba%2Cb%5D" \* MERGEFORMATINET with  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x_1%20%3C%20x_2." \* MERGEFORMATINET By the Mean Value Theorem, there exists  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=c%20%5Cin%20%28x_1%2Cx_2%29" \* MERGEFORMATINET such that  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=f%28x_2%29%20-%20f%28x_1%29%20%3D%20%28x_2%20-%20x_1%29%20f%27%28c%29." \* MERGEFORMATINET  By assumption,  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%28x%29%20%3E%200" \* MERGEFORMATINET for every  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%5Cin%20%5Ba%2Cb%5D" \* MERGEFORMATINET ; hence  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%28c%29%20%3E%200." \* MERGEFORMATINET Also, note that  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x_2%20-%20x_1%20%3E%200." \* MERGEFORMATINET  Hence  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x_2%29%20-%20f%28x_1%29%20%3E%200" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x_2%29%20%3E%20f%28x_1%29." \* MERGEFORMATINET  We can observe the consequences of this theorem by observing the tangent lines of the following graph. Note the tangent lines to the graph, one in each of the intervals  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%280%2Ca%29%2C%20%28a%2Cb%29%2C" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%28b%2C%2B%5Cinfty%29." \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/images?id=123716" \* MERGEFORMATINET  Note first that we have a relative maximum at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20a" \* MERGEFORMATINET and a relative minimum at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20b." \* MERGEFORMATINET The slopes of the tangent lines change from positive for  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%5Cin%20%280%2Ca%29" \* MERGEFORMATINET to negative for  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%5Cin%20%28a%2Cb%29" \* MERGEFORMATINET and then back to positive for  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%5Cin%20%28b%2C%20%2B%5Cinfty%29" \* MERGEFORMATINET . From this we example infer the following theorem: First Derivative Test Suppose that  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET is a continuous function and that  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20c" \* MERGEFORMATINET is a critical value of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f." \* MERGEFORMATINET Then: If  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27" \* MERGEFORMATINET changes from positive to negative at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20c%2C" \* MERGEFORMATINET then  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET has a local maximum at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20c." \* MERGEFORMATINET  If  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27" \* MERGEFORMATINET changes from negative to positive at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20c%2C" \* MERGEFORMATINET then  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET has a local minimum at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20c." \* MERGEFORMATINET  If  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27" \* MERGEFORMATINET does not change sign at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20c%2C" \* MERGEFORMATINET then  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET has neither a local maximum nor minimum at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20c." \* MERGEFORMATINET  Proof of these three conclusions is left to the reader. Example 3: Our previous example showed a graph that had both a local maximum and minimum. Lets reconsider  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%20x%5E3" \* MERGEFORMATINET and observe the graph around  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%200." \* MERGEFORMATINET What happens to the first derivative near this value?  INCLUDEPICTURE "http://www.ck12.org/ck12/images?id=123761" \* MERGEFORMATINET  Example 4: Let's consider the function  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%20%28x%29%20%3D%20x%5E2%20%2B%206%20x%20-%209" \* MERGEFORMATINET and observe the graph around  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20-3." \* MERGEFORMATINET What happens to the first derivative near this value?  INCLUDEPICTURE "http://www.ck12.org/ck12/images?id=123739" \* MERGEFORMATINET  We observe that the slopes of the tangent lines to the graph change from negative to positive at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20-3." \* MERGEFORMATINET The first derivative test verifies this fact. Note that the slopes of the tangent lines to the graph are negative for  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%5Cin%20%28-%5Cinfty%2C%20-3%29" \* MERGEFORMATINET and positive for  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%5Cin%20%28-3%2C%20-%5Cinfty%29." \* MERGEFORMATINET  Lesson Summary We found intervals where a function is increasing and decreasing. We applied the First Derivative Test to find extrema and sketch graphs. Multimedia Links For more examples on determining whether a function is increasing or decreasing (9.0), see  HYPERLINK "http://www.youtube.com/watch?v=6YM3TrudIzQ" Math Video Tutorials by James Sousa, Determining where a function is increasing and decreasing using the first derivative (10:05) INCLUDEPICTURE "http://i3.ytimg.com/vi/6YM3TrudIzQ/hqdefault.jpg" \* MERGEFORMATINET . For a video presentation of increasing and decreasing trigonometric functions and relative extrema (9.0), see  HYPERLINK "http://www.youtube.com/watch?v=NThmBPzQLjU" Math Video Tutorials by James Sousa, Increasing and decreasing trig functions, relative extrema (6:02) INCLUDEPICTURE "http://i3.ytimg.com/vi/NThmBPzQLjU/hqdefault.jpg" \* MERGEFORMATINET . For more information on finding relative extrema using the first derivative (9.0), see  HYPERLINK "http://www.youtube.com/watch?v=Oc6P7p7wyCs" Math Video Tutorials by James Sousa, Finding relative extrema using the first derivative (6:18) INCLUDEPICTURE "http://i4.ytimg.com/vi/Oc6P7p7wyCs/hqdefault.jpg" \* MERGEFORMATINET . Review Questions In problems #12, identify the intervals where the function is increasing, decreasing, or is constant. (Units on the axes indicate single units).  INCLUDEPICTURE "http://www.ck12.org/ck12/images?id=123749" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/images?id=123751" \* MERGEFORMATINET  Give the sign of the following quantities for the graph in #2.  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%28-3%29" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%281%29" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%283%29" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%284%29" \* MERGEFORMATINET  For problems #46, determine the intervals in which the function is increasing and those in which it is decreasing. Sketch the graph.  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%20x%5E2%20-%20%5Cfrac%7B1%7D%20%7Bx%7D" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%20%28x%5E2%20-%201%29%5E5" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%20%28x%5E2%20-%201%29%5E4" \* MERGEFORMATINET  For problems #710, do the following: Use the First Derivative Test to find the intervals where the function increases and/or decreases Identify all absolute and relative max and mins Sketch the graph  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%20-x%5E2%20-4x%20-%201" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%20x%5E3%20%2B%203x%5E2%20-%209x%20%2B%201" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20f%28x%29%20%3D%20x%5E%5Cfrac%7B2%7D%20%7B3%7D%20%28x%20-%205%29" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20f%28x%29%20%3D%202x%7B%5Csqrt%7Bx%5E2%20%2B%201%7D%7D" \* MERGEFORMATINET  Review Answers Increasing on (0, 3), decreasing on (3, 6), constant on (6, "). Increasing on (-", 0) and (3, 7), decreasing on (0, 3).  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%28-3%29%20%3E%200%2C" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%281%29%20%3C%200%2C" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%283%29%20%3D%200%2C" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%284%29%20%3E%200" \* MERGEFORMATINET  Relative minimum at  EMBED Equation.DSMT4 ; increasing on  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 , decreasing on  EMBED Equation.DSMT4 .  INCLUDEPICTURE "http://www.ck12.org/ck12/images?id=123742" \* MERGEFORMATINET  Absolute minimum at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%200" \* MERGEFORMATINET ; decreasing on  EMBED Equation.DSMT4 , increasing on  EMBED Equation.DSMT4   INCLUDEPICTURE "http://www.ck12.org/ck12/images?id=123740" \* MERGEFORMATINET  Absolute minimum at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20%5Cpm%201" \* MERGEFORMATINET ; relative maximum at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%200" \* MERGEFORMATINET ; decreasing on  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%28-%5Cinfty%2C%20-1%29%2C%20%280%2C%201%29%2C" \* MERGEFORMATINET  increasing on  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%28-1%2C%200%29%2C" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%281%2C%20%2B%20%5Cinfty%29." \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/images?id=123732" \* MERGEFORMATINET  Absolute maximum at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20-2" \* MERGEFORMATINET ; increasing on  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%28-%5Cinfty%2C%20-2%29%2C" \* MERGEFORMATINET  decreasing on  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%28-2%2C%20%2B%5Cinfty%29." \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/images?id=123707" \* MERGEFORMATINET  Relative maximum at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20-3%2C" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28-3%29%20%3D%2028" \* MERGEFORMATINET ; relative minimum at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%201%2C" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%281%29%20%3D%20-4" \* MERGEFORMATINET ; increasing on  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%28-%5Cinfty%2C%20-3%29" \* MERGEFORMATINET  and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%281%2C%20%2B%5Cinfty%29%2C" \* MERGEFORMATINET  decreasing on  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%28-3%2C%201%29." \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/images?id=123725" \* MERGEFORMATINET  Relative maximum at x = 0, f(0) = 0; relative minimum at  EMBED Equation.DSMT4 , increasing on  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 , decreasing on (0, 2).  INCLUDEPICTURE "http://www.ck12.org/ck12/images?id=123713" \* MERGEFORMATINET  There are no maxima or minima; no relative maxima or minima.  INCLUDEPICTURE "http://www.ck12.org/ck12/images?id=123762" \* MERGEFORMATINET  5.3 The Second Derivative Test Learning Objectives A student will be able to: Find intervals where a function is concave upward or downward. Apply the Second Derivative Test to determine concavity and sketch graphs. Introduction In this lesson we will discuss a property about the shapes of graphs called concavity, and introduce a method with which to study this phenomenon, the Second Derivative Test. This method will enable us to identify precisely the intervals where a function is either increasing or decreasing, and also help us to sketch the graph. Here is an example that illustrates these properties. Definition A function  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET is said to be concave upward on  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Ba%2C%20b%5D" \* MERGEFORMATINET contained in the domain of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET if  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27" \* MERGEFORMATINET is an increasing function on  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Ba%2C%20b%5D" \* MERGEFORMATINET and concave downward on  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Ba%2C%20b%5D" \* MERGEFORMATINET if  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27" \* MERGEFORMATINET is a decreasing function on  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Ba%2C%20b%5D." \* MERGEFORMATINET  Example 1: Consider the function  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%20x%5E3%20-%20x" \* MERGEFORMATINET :  INCLUDEPICTURE "http://www.ck12.org/ck12/images?id=123712" \* MERGEFORMATINET  The function has zeros at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20%5Cpm%201%2C%200" \* MERGEFORMATINET and has a relative maximum at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20-%5Cfrac%7B%5Csqrt%7B3%7D%7D%20%7B3%7D" \* MERGEFORMATINET and a relative minimum at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20%5Cfrac%7B%5Csqrt%7B3%7D%7D%20%7B3%7D" \* MERGEFORMATINET . Note that the graph appears to be concave down for all intervals in  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%28-%5Cinfty%2C%200%29" \* MERGEFORMATINET and concave up for all intervals in  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%280%2C%20%2B%5Cinfty%29" \* MERGEFORMATINET . Where do you think the concavity of the graph changed from concave down to concave up? If you answered at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%200" \* MERGEFORMATINET you would be correct. In general, we wish to identify both the extrema of a function and also points, the graph changes concavity. The following definition provides a formal characterization of such points. The example above had only one inflection point. But we can easily come up with examples of functions where there are more than one point of inflection. Definition A point on a graph of a function  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET where the concavity changes is called an inflection point. Example 2: Consider the function  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%20x%5E4%20-%203%20x%5E3%20%2B%20x%20-%202." \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/images?id=123753" \* MERGEFORMATINET  We can see that the graph has two relative minimums, one relative maximum, and two inflection points (as indicated by arrows). In general we can use the following two tests for concavity and determining where we have relative maximums, minimums, and inflection points. Test for Concavity Suppose that  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET is continuous on  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Ba%2C%20b%5D" \* MERGEFORMATINET and that  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=I" \* MERGEFORMATINET is some open interval in the domain of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f." \* MERGEFORMATINET  If  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%27%28x%29%20%3E%200" \* MERGEFORMATINET for all  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%5Cin%20I%2C" \* MERGEFORMATINET then the graph of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET is concave upward on  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=I." \* MERGEFORMATINET  If  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%27%28x%29%20%3C%200" \* MERGEFORMATINET for all  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%5Cin%20I%2C" \* MERGEFORMATINET then the graph of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET is concave downward on  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=I." \* MERGEFORMATINET  A consequence of this concavity test is the following test to identify extreme values of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f." \* MERGEFORMATINET  Second Derivative Test for Extrema Suppose that  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET is a continuous function near  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=c" \* MERGEFORMATINET and that  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=c" \* MERGEFORMATINET is a critical value of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f." \* MERGEFORMATINET Then If  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%27%28c%29%20%3E%200%2C" \* MERGEFORMATINET then  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET has a relative maximum at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20c." \* MERGEFORMATINET  If  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%27%28c%29%20%3C%200%2C" \* MERGEFORMATINET then  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET has a relative minimum at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20c." \* MERGEFORMATINET  If  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%27%28c%29%20%3D%200%2C" \* MERGEFORMATINET then the test is inconclusive and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20c" \* MERGEFORMATINET may be a point of inflection. Recall the graph  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%20x%5E3." \* MERGEFORMATINET We observed that  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%200%2C" \* MERGEFORMATINET and that there was neither a maximum nor minimum. The Second Derivative Test cautions us that this may be the case since at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%27%280%29%20%3D%200" \* MERGEFORMATINET at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%200." \* MERGEFORMATINET  So now we wish to use all that we have learned from the First and Second Derivative Tests to sketch graphs of functions. The following table provides a summary of the tests and can be a useful guide in sketching graphs. Signs of first and second derivatives Information from applying First and Second Derivative Tests Shape of the graphs  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%28x%29%20%3E%200" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%27%28x%29%20%3E%200" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET is increasing  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET is concave upward  INCLUDEPICTURE "http://www.ck12.org/ck12/images?id=123734" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%28x%29%20%3E%200" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%27%28x%29%20%3C%200" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET is increasing  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET is concave downward  INCLUDEPICTURE "http://www.ck12.org/ck12/images?id=123763" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%28x%29%20%3C%200" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%27%28x%29%20%3E%200" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET is decreasing  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET is concave upward  INCLUDEPICTURE "http://www.ck12.org/ck12/images?id=123720" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%28x%29%20%3C%200" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%27%28x%29%20%3C%200" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET is decreasing  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET is concave downward  INCLUDEPICTURE "http://www.ck12.org/ck12/images?id=123731" \* MERGEFORMATINET Lets look at an example where we can use both the First and Second Derivative Tests to find out information that will enable us to sketch the graph. Example 3: Lets examine the function  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%20x%5E5%20-%205x%20%2B%202." \* MERGEFORMATINET  1. Find the critical values for which  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%28c%29%20%3D%200." \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%28x%29%20%3D%205x%5E4%20-%205%20%3D%200%2C" \* MERGEFORMATINET or  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%5E4%20-%201%20%3D%200" \* MERGEFORMATINET at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20%5Cpm%201." \* MERGEFORMATINET  Note that  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%27%28x%29%20%3D%2020%20x%5E3%20%3D%200" \* MERGEFORMATINET when  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%200." \* MERGEFORMATINET  2. Apply the First and Second Derivative Tests to determine extrema and points of inflection. We can note the signs of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%27" \* MERGEFORMATINET in the intervals partitioned by  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20%5Cpm%201%2C%200." \* MERGEFORMATINET  Key intervals INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%28x%29" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%27%28x%29" \* MERGEFORMATINET Shape of graph INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3C%20-1" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%2B" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=-" \* MERGEFORMATINET Increasing, concave down  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=-1%20%3C%20%5Ctimes%20%3C%200" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=-" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=-" \* MERGEFORMATINET Decreasing, concave down  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=0%20%3C%20%5Ctimes%20%3C%201" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=-" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%2B" \* MERGEFORMATINET Decreasing, concave up  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3E%201" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%2B" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%2B" \* MERGEFORMATINET Increasing, concave up Also note that  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%27%20%28-1%29%20%3D%20-%2020%20%3C%200." \* MERGEFORMATINET By the Second Derivative Test we have a relative maximum at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20-1%2C" \* MERGEFORMATINET  or the point  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%28-1%2C%206%29." \* MERGEFORMATINET  In addition,  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%27%20%281%29%20%3D%2020%20%3E%200." \* MERGEFORMATINET By the Second Derivative Test we have a relative minimum at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%201%2C" \* MERGEFORMATINET or the point  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%281%2C%20-2%29." \* MERGEFORMATINET Now we can sketch the graph.  INCLUDEPICTURE "http://www.ck12.org/ck12/images?id=123773" \* MERGEFORMATINET  Lesson Summary We learned to identify intervals where a function is concave upward or downward. We applied the First and Second Derivative Tests to determine concavity and sketch graphs. Multimedia Links For a video presentation of the second derivative test to determine relative extrema (9.0), see  HYPERLINK "http://www.youtube.com/watch?v=2tmRPytHBuk" Math Video Tutorials by James Sousa, Introduction to Limits (8:46) INCLUDEPICTURE "http://i3.ytimg.com/vi/2tmRPytHBuk/hqdefault.jpg" \* MERGEFORMATINET . Review Questions Find all extrema using the Second Derivative Test.  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20f%28x%29%20%3D%5Cfrac%7Bx%5E2%7D%20%7B4%7D%20%2B%20%5Cfrac%7B4%7D%20%7Bx%7D" \* MERGEFORMATINET  Consider  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%20x%5E2%20%2B%20ax%20%2B%20b%2C" \* MERGEFORMATINET  with  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%281%29%20%3D%203." \* MERGEFORMATINET  Determine  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=a" \* MERGEFORMATINET  and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=b" \* MERGEFORMATINET  so that  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%201" \* MERGEFORMATINET  is a critical value of the function f. Is the point (1, 3) a maximum, a minimum, or neither? In problems #36, find all extrema and inflection points. Sketch the graph.  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%20x%5E3%20%2B%20x%5E2" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20f%28x%29%20%3D%20%5Cfrac%7Bx%5E2%20%2B3%7D%20%7Bx%7D" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%20x%5E3%20-%2012x" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%20-%5Cfrac%7B1%7D%20%7B4%7D%20x%5E4%20%2B%202x%5E2" \* MERGEFORMATINET  Use your graphing calculator to examine the graph of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%20x%28x%20-%201%29%5E3" \* MERGEFORMATINET  (Hint: you will need to change the y range in the viewing window) Discuss the concavity of the graph in the interval  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%280%2C%20%5Cfrac%7B1%7D%20%7B2%7D%29" \* MERGEFORMATINET . Use your calculator to find the minimum value of the function in that interval. True or False:  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%20x%5E4%20%2B%204x%5E3" \* MERGEFORMATINET  has a relative minimum at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20-2" \* MERGEFORMATINET  and a relative maximum at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%200" \* MERGEFORMATINET . If possible, provide an example of a non-polynomial function that has exactly one relative minimum. If possible, provide an example of a non-polynomial function that is concave downward everywhere in its domain. Review Answers There is a relative minimum at x = 2, and it is located at (2, 3). f(1) = 3 suggests that  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 ; solving this system we have that a = -2 and b = 4. The point (1, 3) is an absolute min of f. Relative maximum at  EMBED Equation.DSMT4 , relative minimum at x = 0; the relative maximum is located at  EMBED Equation.DSMT4 ; the relative minimum is located at (0, 0). There is a point of inflection at  EMBED Equation.DSMT4 .  INCLUDEPICTURE "http://www.ck12.org/ck12/images?id=123744" \* MERGEFORMATINET  Relative maximum at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20-%5Csqrt%7B3%7D" \* MERGEFORMATINET , located at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%28-%5Csqrt%7B3%7D%2C%20-2%5Csqrt%7B3%7D%29" \* MERGEFORMATINET ; relative minimum at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20%7B%5Csqrt%7B3%7D%7D" \* MERGEFORMATINET , located at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%28%5Csqrt%7B3%7D%2C%202%7B%5Csqrt%7B3%7D%7D%29" \* MERGEFORMATINET . There are no inflection points.  INCLUDEPICTURE "http://www.ck12.org/ck12/images?id=123738" \* MERGEFORMATINET  Relative maximum at x = -2; relative minimum at x = 2; the relative maximum is located at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%28-2%2C%2016%29" \* MERGEFORMATINET ; the relative minimum is located at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%282%2C%2016%29." \* MERGEFORMATINET  There is a point of inflection at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%280%2C%200%29." \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/images?id=123754" \* MERGEFORMATINET  Relative maxima at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20%5Cpm%202%2C" \* MERGEFORMATINET  relative minimum at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%200" \* MERGEFORMATINET ; the relative maxima are located at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%28-2%2C%204%29" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%282%2C%204%29" \* MERGEFORMATINET ; the relative minimum is located at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%280%2C%200%29." \* MERGEFORMATINET  There are two inflection points, located at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%28-%20%5Cfrac%7B2%20%5Csqrt%7B3%7D%7D%20%7B3%7D%2C%20%5Cfrac%7B20%7D%20%7B9%7D%29" \* MERGEFORMATINET  and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%28%5Cfrac%7B2%20%5Csqrt%7B3%7D%7D%20%7B3%7D%2C%20%5Cfrac%7B20%7D%20%7B9%7D%29." \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/images?id=123772" \* MERGEFORMATINET  The graph is concave up in the interval. There is a relative minimum at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%280.25%2C%20-0.10%29" \* MERGEFORMATINET  False: there are inflection points at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%200" \* MERGEFORMATINET  and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20-2" \* MERGEFORMATINET . There is a relative minimum at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20-3." \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/images?id=123757" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%20x%5E2%20%2B%20%28%5Cfrac%7B1%7D%20%7Bx%20-%201%7D%29" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/images?id=123719" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%20%7B%5Csqrt%7Bx%7D%7D" \* MERGEFORMATINET  on  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%280%2C%20%2B%20%5Cinfty%29" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/images?id=123728" \* MERGEFORMATINET  Also,  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%20-%20%5Cfrac%7B1%7D%20%7Bx%5E2%7D" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/images?id=123756" \* MERGEFORMATINET  Second Derivative Test Practice State the intervals on which the function is concave up and concave down and state all points of inflection. 1.)  EMBED Equation.DSMT4  2.)  EMBED Equation.DSMT4  3.)  EMBED Equation.DSMT4  4.)  EMBED Equation.DSMT4  5.)  EMBED Equation.DSMT4  6.)  EMBED Equation.DSMT4  7.)  EMBED Equation.DSMT4  Answers: 1234567CU EMBED Equation.DSMT4  EMBED Equation.DSMT4  EMBED Equation.DSMT4  EMBED Equation.DSMT4  EMBED Equation.DSMT4  EMBED Equation.DSMT4  EMBED Equation.DSMT4 CDnowhere EMBED Equation.DSMT4  EMBED Equation.DSMT4 nowhere EMBED Equation.DSMT4  EMBED Equation.DSMT4  EMBED Equation.DSMT4 POInone EMBED Equation.DSMT4 (3, 0)none EMBED Equation.DSMT4  EMBED Equation.DSMT4  EMBED Equation.DSMT4  Concavity HW Determine intervals where each function is concave up and concave down, and find all inflection points for each function. 1.)  EMBED Equation.DSMT4  2.)  EMBED Equation.DSMT4  3.)  EMBED Equation.DSMT4  4.)  EMBED Equation.DSMT4  5.)  EMBED Equation.DSMT4  6.)  EMBED Equation.DSMT4  7.)  EMBED Equation.DSMT4  Answers! 1234567C.Up EMBED Equation.DSMT4  EMBED Equation.DSMT4  EMBED Equation.DSMT4  EMBED Equation.DSMT4  EMBED Equation.DSMT4  EMBED Equation.DSMT4  EMBED Equation.DSMT4 C.Downnowhere EMBED Equation.DSMT4  EMBED Equation.DSMT4 nowhere EMBED Equation.DSMT4  EMBED Equation.DSMT4  EMBED Equation.DSMT4 IPnone EMBED Equation.DSMT4 (3, 0)none EMBED Equation.DSMT4  EMBED Equation.DSMT4  EMBED Equation.DSMT4  Curve Sketching Practice For each function, use the First and Second Derivative Tests to find the intervals where the function is increasing/decreasing/concave up/concave down, all extrema, and all points of inflection. Then, use that information to sketch the graph, labeling the important points. Feel free to use your calculator to check your sketches. 1.)  EMBED Equation.DSMT4  2.)  EMBED Equation.DSMT4  3.)  EMBED Equation.DSMT4  4.)  EMBED Equation.DSMT4  Big Problem HW For each function, find intervals of increasing / decreasing / concave up / concave down, all extrema, and all inflection points. (Do this without using your calculator.) Then, use that information to sketch the graph. Feel free to use your calculator to check your sketches. 1.)  EMBED Equation.DSMT4  2.)  EMBED Equation.DSMT4  3.)  EMBED Equation.DSMT4  4.)  EMBED Equation.DSMT4  5.4 Limits at Infinity Learning Objectives A student will be able to: Examine end behavior of functions on infinite intervals. Determine horizontal asymptotes. Examine indeterminate forms of limits of rational functions. Apply LHospitals Rule to find limits. Examine infinite limits at infinity. Introduction In this lesson we will return to the topics of infinite limits and end behavior of functions and introduce a new method that we can use to determine limits that have indeterminate forms. Examine End Behavior of Functions on Infinite Intervals Suppose we are trying to analyze the end behavior of rational functions. In Lesson on Infinite Limits we looked at some rational functions such as  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%20%5Cfrac%7B2x%5E2%20-%201%7D%20%7Bx%5E2%20-%201%7D" \* MERGEFORMATINET and showed that  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Clim_%7Bx%20%5Cto%20%2B%5Cinfty%7D%20f%28x%29%3D%202" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Clim_%7Bx%20%5Cto%20-%5Cinfty%7D%20f%28x%29%3D%202" \* MERGEFORMATINET . We required an analysis of the end behavior of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET since computing the limit by direct substitution yielded the indeterminate form  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cfrac%20%7B%5Cinfty%7D%20%7B%5Cinfty%7D" \* MERGEFORMATINET . Our approach to compute the infinite limit was to look at actual values of the function  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29" \* MERGEFORMATINET as  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x" \* MERGEFORMATINET approached  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cpm%20%5Cinfty" \* MERGEFORMATINET . We interpreted the result graphically as the function having a horizontal asymptote at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%202." \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/images?id=123737" \* MERGEFORMATINET  We were then able to find infinite limits of more complicated rational functions such as  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Clim_%7Bx%20%5Cto%20%5Cinfty%7D%20%5Cfrac%7B3x%5E4%20-%202x%5E2%20%2B%203x%20%2B%201%7D%20%7B2x%5E4%20-%202x%5E2%20%2B%20x%20-%203%7D%20%3D%20%5Cfrac%7B3%7D%20%7B2%7D" \* MERGEFORMATINET using the fact that  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Clim_%7Bx%20%5Cto%20%5Cinfty%7D%20%5Cfrac%7B1%7D%20%7Bx%5Ep%7D%20%3D%20%5Ctheta%2C%20p%20%3E%20%5Ctheta" \* MERGEFORMATINET . Similarly, we used such an approach to compute limits whenever direct substitution resulted in the indeterminate form  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7B%5Ctheta%7D%7B%5Ctheta%7D" \* MERGEFORMATINET , such as  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Clim_%7Bx%5Cto%201%7D%20%5Cfrac%7Bx%5E2%20-%201%7D%20%7Bx%20-%201%7D%20%3D%202" \* MERGEFORMATINET . Now lets consider other functions of the form  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%28f%28x%29/g%28x%29%29" \* MERGEFORMATINET where we get the indeterminate forms  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7B%5Ctheta%7D%20%7B%5Ctheta%7D" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7B%5Ctheta%7D%20%7B%5Ctheta%7D" \* MERGEFORMATINET and determine an appropriate analytical method for computing the limits. Example 1: Consider the function  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%20%5Cfrac%7B%7B%5Cln%7D%28x%20%2B1%29%7D%20%7Bx%7D" \* MERGEFORMATINET and suppose we wish to find  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Clim_%7Bx%5Cto%200%7D%20%5Cfrac%7B%7B%5Cln%7D%28x%20%2B%201%29%7D%20%7Bx%7D" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Clim_%7Bx%20%5Cto%20%5Cinfty%7D%20%5Cfrac%7B%7B%5Cln%7D%28x%20%2B%201%29%7D%20%7Bx%7D" \* MERGEFORMATINET We note the following: Direct substitution leads to the indeterminate forms  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7B0%7D%20%7B0%7D" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7B%5Cinfty%7D%20%7B%5Cinfty%7D." \* MERGEFORMATINET  The function in the numerator is not a polynomial function, so we cannot use our previous methods such as applying  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Clim_%7Bx%20%5Cto%20%5Cinfty%7D%20%5Cfrac%7B1%7D%20%7Bx%5Ep%7D%20%3D%20%5Ctheta." \* MERGEFORMATINET  Lets examine both the graph and values of the function for appropriate  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x" \* MERGEFORMATINET values, to see if they cluster around particular  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=y" \* MERGEFORMATINET values. Here is a sketch of the graph and a table of extreme values. We first note that domain of the function is  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%28-1%2C%200%29%20%5Ccup%20%280%2C%20%2B%5Cinfty%29" \* MERGEFORMATINET and is indicated in the graph as follows:  INCLUDEPICTURE "http://www.ck12.org/ck12/images?id=123736" \* MERGEFORMATINET  So,  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Clim_%7Bx%5Cto%200%7D%20%5Cfrac%7B%7B%5Cln%7D%28x%20%2B%201%29%7D%20%7Bx%7D" \* MERGEFORMATINET appears to approach the value  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=1" \* MERGEFORMATINET as the following table suggests. Note: Please see Differentiation and Integration of Logarithmic and Exponential Functions in Chapter 6 for more on derivatives of Logarithmic functions.  INCLUDEPICTURE "http://www.ck12.org/ck12/images?id=123743" \* MERGEFORMATINET   So we infer that  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Clim_%7Bx%5Cto%200%7D%20%5Cfrac%7B%7B%5Cln%7D%28x%20%2B%201%29%7D%20%7Bx%7D%20%3D%201" \* MERGEFORMATINET . For the infinite limit,  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Clim_%7Bx%20%5Cto%20%5Cinfty%7D%20%5Cfrac%7B%7B%5Cln%7D%28x%20%2B%201%29%7D%20%7Bx%7D%20%3D%201" \* MERGEFORMATINET , the inference of the limit is not as obvious. The function appears to approach the value  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=0" \* MERGEFORMATINET  but does so very slowly, as the following table suggests.  This unpredictable situation will apply to many other functions of the form. Hence we need another method that will provide a different tool for analyzing functions of the form  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7Bf%28x%29%7D%20%7Bg%28x%29%7D" \* MERGEFORMATINET . LHospitals Rule: Let functions  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=g" \* MERGEFORMATINET be differentiable at every number other than  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=c" \* MERGEFORMATINET in some interval, with  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=g%27%20%28x%29%20%5Cneq%200" \* MERGEFORMATINET if  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%5Cneq%20c." \* MERGEFORMATINET If  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Clim_%7Bx%5Cto%20c%7D%20f%28x%29%20%3D%20%5Clim_%7Bx%5Cto%20c%7D%20g%28x%29%20%3D%200" \* MERGEFORMATINET , or if  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Clim_%7Bx%5Cto%20c%7D%20f%28x%29%20%3D%20%5Cpm%20%5Cinfty" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Clim_%7Bx%5Cto%20c%7D%20g%28x%29%20%3D%20%5Cpm%20%5Cinfty%2C" \* MERGEFORMATINET then:  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Clim_%7Bx%5Cto%20c%7D%20%5Cfrac%7Bf%28x%29%7D%20%7Bg%28x%29%7D%20%3D%20%5Clim_%7Bx%5Cto%20c%7D%20%5Cfrac%7Bf%27%28x%29%7D%20%7Bg%27%28x%29%7D" \* MERGEFORMATINET as long as this latter limit exists or is infinite. If  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=g" \* MERGEFORMATINET are differentiable at every number  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x" \* MERGEFORMATINET greater than some number  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=a" \* MERGEFORMATINET , with  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=g%27%28x%29%20%5Cneq%200%2C" \* MERGEFORMATINET then  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Clim_%7Bx%20%5Cto%20%5Cinfty%7D%20%5Cfrac%7Bf%28x%29%7D%20%7Bg%28x%29%7D%20%3D%20%5Clim_%7Bx%20%5Cto%20%5Cinfty%7D%20%5Cfrac%7Bf%27%28x%29%7D%20%7Bg%27%28x%29%7D" \* MERGEFORMATINET as long as this latter limit exists or is infinite. Lets look at applying the rule to some examples. Example 2: We will start by reconsidering the previous example,  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%20%5Cfrac%7B%7B%5Cln%7D%28x%20%2B1%29%7D%20%7Bx%7D%2C" \* MERGEFORMATINET and verify the following limits using LHospitals Rule:  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%5Clim_%7Bx%5Cto%200%7D%20%5Cfrac%7B%7B%5Cln%7D%28x%20%2B%201%29%7D%20%7Bx%7D%20%26%20%3D%201.%5C%5C%0A%5Clim_%7Bx%20%5Cto%20%5Cinfty%7D%20%5Cfrac%7B%7B%5Cln%7D%28x%20%2B%201%29%7D%20%7Bx%7D%20%26%20%3D%200." \* MERGEFORMATINET  Solution: Since  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Clim_%7Bx%5Cto%200%7D%20%7B%5Cln%7D%28x%20%2B%201%29%20%3D%20%5Clim_%7Bx%5Cto%200%7D%20x%20%3D%200" \* MERGEFORMATINET , LHospitals Rule applies and we have  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%5Clim_%7Bx%5Cto%200%7D%20%5Cfrac%7B%7B%5Cln%7D%28x%20%2B%201%29%7D%20%7Bx%7D%20%3D%20%5Clim_%7Bx%5Cto%200%7D%20%5Cfrac%7B%5Cfrac%7B1%7D%20%7Bx%20%2B%201%7D%7D%20%7B1%7D%20%3D%20%5Cfrac%7B1%7D%20%7B1%7D%20%3D%201." \* MERGEFORMATINET  Likewise,  Now lets look at some more examples. Example 3: Evaluate  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Clim_%7Bx%5Cto%200%7D%20%5Cfrac%7Be%5Ex%20-%201%7D%20%7Bx%7D." \* MERGEFORMATINET  Solution: Since  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Clim_%7Bx%5Cto%200%7D%20%28e%5Ex%20-%201%29%20%3D%20%5Clim_%7Bx%5Cto%200%7D%20x%20%3D0" \* MERGEFORMATINET , LHospitals Rule applies and we have  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%5Clim_%7Bx%5Cto%200%7D%20%5Cfrac%7Be%5Ex%20-%201%7D%20%7Bx%7D%20%3D%20%5Clim_%7Bx%5Cto%200%7D%20%5Cfrac%7Be%5Ex%7D%20%7B1%7D%20%3D%20%5Cfrac%7B1%7D%20%7B1%7D%20%3D%201." \* MERGEFORMATINET  Lets look at an example with trigonometric functions. Example 4: Evaluate  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Clim_%7Bx%5Cto%200%7D%20%5Cfrac%7B1%20-%20%5Ccos%20x%7D%7Bx%5E2%7D." \* MERGEFORMATINET  Solution: Since  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Clim_%7Bx%5Cto%200%7D%20%281%20-%20%5Ccos%20x%29%20%3D%20%5Clim_%7Bx%5Cto%200%7D%20x%5E2%20%3D%200" \* MERGEFORMATINET , LHospitals Rule applies and we have  Example 5: Evaluate  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Clim_%7Bx%20%5Cto%20%2B%5Cinfty%7D%20%5Cfrac%7Bx%5E2%7D%20%7Be%5Ex%7D" \* MERGEFORMATINET  Solution: Since  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Clim_%7Bx%20%5Cto%20%2B%5Cinfty%7D%20x%5E2%20%3D%20%5Clim_%7Bx%20%5Cto%20%2B%5Cinfty%7D%20e%5Ex%20%3D%20%2B%20%5Cinfty" \* MERGEFORMATINET , LHospitals Rule applies and we have  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%5Clim_%7Bx%20%5Cto%20%2B%5Cinfty%7D%20%5Cfrac%7Bx%5E2%7D%20%7Be%5Ex%7D%20%3D%20%5Clim_%7Bx%20%5Cto%20%2B%5Cinfty%7D%20%5Cfrac%7B2x%7D%20%7Be%5Ex%7D." \* MERGEFORMATINET  Here we observe that we still have the indeterminate form  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7B%5Cinfty%7D%20%7B%5Cinfty%7D" \* MERGEFORMATINET . So we apply LHospitals Rule again to find the limit as follows:  L'Hospital's Rule can be used repeatedly on functions like this. It is often useful because polynomial functions can be reduced to a constant. Lesson Summary We learned to examine end behavior of functions on infinite intervals. We determined horizontal asymptotes of rational functions. We examined indeterminate forms of limits of rational functions. We applied LHospitals Rule to find limits of rational functions. We examined infinite limits at infinity. Review Questions Use your graphing calculator to estimate  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Clim_%7Bx%20%5Cto%20%2B%5Cinfty%7D%20%5Cleft%20%5Bx%5B%7B%5Cln%7D%28x%20%2B%203%29%20-%20%7B%5Cln%7D%28x%29%5D%5Cright%5D." \* MERGEFORMATINET  Use your graphing calculator to estimate  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Clim_%7Bx%20%5Cto%20%2B%5Cinfty%7D%20%5Cfrac%7Bx%7D%20%7B%7B%5Cln%7D%281%20%2B%202e%5Ex%29%7D." \* MERGEFORMATINET  In problems #310, use LHospitals Rule to compute the limits, if they exist.  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Clim_%7Bx%5Cto%203%7D%20%5Cfrac%7Bx%5E2%20-9%7D%20%7Bx%20-%203%7D" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Clim_%7Bx%5Cto%200%7D%20%5Cfrac%7B%5Csqrt%7B1%20%2B%20x%7D%20-%20%5Csqrt%7B1%20-%20x%7D%7D%20%7Bx%7D" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Clim_%7Bx%20%5Cto%20%2B%5Cinfty%7D%20%5Cfrac%7B%7B%5Cln%7D%28x%29%7D%20%7B%5Csqrt%7Bx%7D%7D" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Clim_%7Bx%20%5Cto%20%2B%5Cinfty%7D%20x%5E2%20e%5E%7B-2x%7D" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Clim_%7Bx%5Cto%200%7D%20%281%20-%20x%29%5E%5Cfrac%7B1%7D%7Bx%7D" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Clim_%7Bx%5Cto%200%7D%20%5Cfrac%7Be%5Ex%20-1%20-x%7D%20%7Bx%5E2%7D" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Clim_%7Bx%20%5Cto%20-%5Cinfty%7D%20%5Cfrac%7Be%5Ex%20-1%20-x%7D%20%7Bx%5E2%7D" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Clim_%7Bx%20%5Cto%20%5Cinfty%7D%20x%5E%5Cfrac%7B1%7D%20%7B4%7D%20%7B%5Cln%7D%28x%29" \* MERGEFORMATINET  Review Answers  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Clim_%7Bx%20%5Cto%20%2B%5Cinfty%7D%20%5Cleft%5Bx%5B%7B%5Cln%7D%28x%20%2B%203%29%20-%20%7B%5Cln%7D%20%28x%29%5D%5Cright%5D%20%3D%203" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Clim_%7Bx%20%5Cto%20%2B%5Cinfty%7D%20%5Cfrac%7Bx%7D%20%7B%7B%5Cln%7D%281%20%2B%202e%5Ex%29%7D%20%3D%201" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Clim_%7Bx%5Cto%203%7D%20%5Cfrac%7Bx%5E2%20-9%7D%20%7Bx%20-%203%7D%20%3D%206" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Clim_%7Bx%5Cto%200%7D%20%5Cfrac%7B%5Csqrt%7B1%20%2B%20x%7D%20-%20%5Csqrt%7B1%20-%20x%7D%7D%20%7Bx%7D%20%3D%201" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Clim_%7Bx%20%5Cto%20%2B%5Cinfty%7D%20%5Cfrac%7B%7B%5Cln%7D%20%28x%29%7D%20%7B%5Csqrt%7Bx%7D%7D%20%3D%200" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Clim_%7Bx%20%5Cto%20%2B%5Cinfty%7D%20x%5E2%20e%5E%7B-2x%7D%20%3D%200" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Clim_%7Bx%5Cto%200%7D%20%281%20-%20x%29%5E%7B%5Cfrac%7B1%7D%20%7Bx%7D%7D%20%3D%20%5Cfrac%7B1%7D%20%7Be%7D" \* MERGEFORMATINET  Hint: Let  EMBED Equation.DSMT4 , so  EMBED Equation.DSMT4   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Clim_%7Bx%5Cto%200%7D%20%5Cfrac%7Be%5Ex%20-1%20-x%7D%20%7Bx%5E2%7D%20%3D%20%5Cfrac%7B1%7D%7B2%7D" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Clim_%7Bx%20%5Cto%20-%5Cinfty%7D%20%5Cfrac%7Be%5Ex%20-1%20-x%7D%20%7Bx%5E2%7D%20%3D%200" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Clim_%7Bx%20%5Cto%20%5Cinfty%7D%20x%5E%5Cfrac%7B1%7D%20%7B4%7D%20%7B%5Cln%7D%20%28x%29%20%3D%200" \* MERGEFORMATINET  Asymptote & Limits Review HW For each function, find all requested info, if possible. 1.)  EMBED Equation.DSMT4  Roots: Holes: VA: y-int:  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  2.)  EMBED Equation.DSMT4  Roots: Holes: VA: y-int:  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  3.)  EMBED Equation.DSMT4  Roots: Holes: VA: y-int:  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  4.)  EMBED Equation.DSMT4  Roots: Holes: VA: y-int:  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  5.)  EMBED Equation.DSMT4  Roots: Holes: VA: y-int:  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  6.)  EMBED Equation.DSMT4  Roots: Holes: VA: y-int:  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   Answers: RootsHolesVAy-int EMBED Equation.DSMT4  EMBED Equation.DSMT4  EMBED Equation.DSMT4 1.)nonenonex = 0none1 EMBED Equation.DSMT4  EMBED Equation.DSMT4 2.) EMBED Equation.DSMT4 nonex = 2, x = -1(0, 1)12:  EMBED Equation.DSMT4 ; -1:  EMBED Equation.DSMT4 2:  EMBED Equation.DSMT4 ; -1:  EMBED Equation.DSMT4 3.) EMBED Equation.DSMT4 none EMBED Equation.DSMT4  EMBED Equation.DSMT4  EMBED Equation.DSMT4 2:  EMBED Equation.DSMT4 ; -2:  EMBED Equation.DSMT4 2:  EMBED Equation.DSMT4 ; -2:  EMBED Equation.DSMT4 4.)(4, 0) EMBED Equation.DSMT4 x = 3 EMBED Equation.DSMT4 1 EMBED Equation.DSMT4  EMBED Equation.DSMT4 5.)(0, 0)none EMBED Equation.DSMT4 (0, 0)0 EMBED Equation.DSMT4  EMBED Equation.DSMT4 6.)nonenonex = 2 EMBED Equation.DSMT4 0 EMBED Equation.DSMT4  EMBED Equation.DSMT4  5.5 Analyzing the Graph of a Function Learning Objectives A student will be able to: Summarize the properties of function including intercepts, domain, range, continuity, asymptotes, relative extreme, concavity, points of inflection, limits at infinity. Apply the First and Second Derivative Tests to sketch graphs. Introduction In this lesson we summarize what we have learned about using derivatives to analyze the graphs of functions. We will demonstrate how these various methods can be applied to help us examine a functions behavior and sketch its graph. Since we have already discussed the various techniques, this lesson will provide examples of using the techniques to analyze the examples of representative functions we introduced in the Lesson on Relations and Functions, particularly rational, polynomial, radical, and trigonometric functions. Before we begin our work on these examples, it may be useful to summarize the kind of information about functions we now can generate based on our previous discussions. Let's summarize our results in a table like the one shown because it provides a useful template with which to organize our findings. Table Summary INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29" \* MERGEFORMATINET Analysis Domain and RangeIntercepts and ZerosAsymptotes and limits at infinityDifferentiabilityIntervals where  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET is increasingIntervals where  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET is decreasingRelative extremaConcavityInflection pointsExample 1: Analyzing Rational Functions Consider the function  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%20%5Cfrac%7Bx%5E2%20-4%7D%20%7Bx%5E2%20-%202x%20-%208%7D." \* MERGEFORMATINET  General Properties: The function appears to have zeros at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20%5Cpm%202." \* MERGEFORMATINET However, once we factor the expression we see  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%5Cfrac%7Bx%5E2%20-%204%7D%20%7Bx%5E2%20-2x%20-8%7D%20%3D%20%5Cfrac%7B%28x%20%2B%202%29%28x%20-%202%29%7D%20%7B%28x%20-%204%29%28x%20%2B%202%29%7D%20%3D%20%5Cfrac%7Bx%20-%202%7D%20%7Bx%20-%204%7D." \* MERGEFORMATINET  Hence, the function has a zero at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%202%2C" \* MERGEFORMATINET there is a hole in the graph at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20-2%2C" \* MERGEFORMATINET the domain is  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%28-%5Cinfty%2C%20-2%29%20%5Ccup%20%28-2%2C%204%29%20%5Ccup%20%284%2C%20%2B%5Cinfty%29%2C" \* MERGEFORMATINET and the  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=y-" \* MERGEFORMATINET intercept is at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%280%2C%20%5Cfrac%7B1%7D%20%7B2%7D%29." \* MERGEFORMATINET  Asymptotes and Limits at Infinity Given the domain, we note that there is a vertical asymptote at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%204." \* MERGEFORMATINET To determine other asymptotes, we examine the limit of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET as  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20x%20%5Crightarrow%20%5Cinfty" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20x%20%5Crightarrow%20-%5Cinfty" \* MERGEFORMATINET . We have  Similarly, we see that  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Clim_%7Bx%20%5Cto%20-%5Cinfty%7D%20%5Cfrac%7Bx%5E2%20-4%7D%20%7Bx%5E2%20-2x%20-%208%7D%20%3D1" \* MERGEFORMATINET . We also note that  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=y%20%5Cneq%201" \* MERGEFORMATINET since  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%5Cneq%20-2." \* MERGEFORMATINET  Hence we have a horizontal asymptote at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=y%20%3D%201." \* MERGEFORMATINET  Differentiability  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%28x%29%20%3D%20%5Cfrac%7B-2x%5E2%20-8x%20-%208%7D%20%7B%28x%5E2%20-2x%20-%208%29%7D%20%3D%20%5Cfrac%7B-2%7D%20%7B%28x%20-%204%29%5E2%7D%20%3C0" \* MERGEFORMATINET . Hence the function is differentiable at every point of its domain, and since  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%28x%29%20%3C%200" \* MERGEFORMATINET on its domain, then  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET is decreasing on its domain,  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%28-%5Cinfty%2C%20-2%29%20%5Ccup%20%28-2%2C%204%29%20%5Ccup%20%284%2C%20%2B%5Cinfty%29" \* MERGEFORMATINET .  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%20f%27%27%28x%29%20%3D%20%5Cfrac%7B4%7D%20%7B%28x%20-%204%29%5E3%7D." \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%27%28x%29%20%5Cneq%200" \* MERGEFORMATINET in the domain of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f." \* MERGEFORMATINET Hence there are no relative extrema and no inflection points. So  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%27%28x%29%20%3E%200" \* MERGEFORMATINET when  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3E%204." \* MERGEFORMATINET Hence the graph is concave up for  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3E%204." \* MERGEFORMATINET  Similarly,  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%27%28x%29%20%3C%200" \* MERGEFORMATINET when  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3C%204." \* MERGEFORMATINET Hence the graph is concave down for  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3C%204%2C" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%5Cneq%20-2." \* MERGEFORMATINET  Lets summarize our results in the table before we sketch the graph. Table Summary INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%3D%5Cfrac%7Bx%5E2%20-4%7D%7Bx%5E2-2x-8%7D" \* MERGEFORMATINET AnalysisDomain and Range  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=D%3D%28-%5Cinfty%2C-2%29%5Ccup%28-2%2C4%29%5Ccup%284%2C%2B%5Cinfty%29" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=R%3D%5Cbig%5C%7B%5Cmathrm%7Ball%5C%20reals%7D%5Cneq%201%5Cbig%5C%7D" \* MERGEFORMATINET Intercepts and Zeroszero at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%202%2C" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=y-" \* MERGEFORMATINET intercept at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cleft%280%2C%5Cfrac%7B1%7D%7B2%7D%20%5Cright%29" \* MERGEFORMATINET Asymptotes and limits at infinityVA at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%204%2C" \* MERGEFORMATINET HA at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=y%20%3D%201%2C" \* MERGEFORMATINET hole in the graph at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20-2" \* MERGEFORMATINET Differentiabilitydifferentiable at every point of its domain Intervals where  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET is increasingnowhere Intervals where  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET is decreasing INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%28-%5Cinfty%2C-2%29%5Ccup%28-2%2C4%29%5Ccup%284%2C%2B%5Cinfty%29" \* MERGEFORMATINET Relative extremanone Concavityconcave up in  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%284%2C%2B%5Cinfty%29%2C" \* MERGEFORMATINET concave down in  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%28-%5Cinfty%2C-2%29%5Ccup%28-2%2C4%29" \* MERGEFORMATINET Inflection pointsnone Finally, we sketch the graph as follows:  INCLUDEPICTURE "http://www.ck12.org/ck12/images?id=123723" \* MERGEFORMATINET  Lets look at examples of the other representative functions we introduced in Lesson 1.2. Example 2: Analyzing Polynomial Functions Consider the function  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%20x%5E3%20%2B%202%20x%5E2%20-%20x%20-%202." \* MERGEFORMATINET  General Properties The domain of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET is  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%28-%5Cinfty%2C%2B%5Cinfty%29" \* MERGEFORMATINET and the  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=y-" \* MERGEFORMATINET intercept at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%280%2C-2%29." \* MERGEFORMATINET  The function can be factored  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=f%28x%29%20%3D%20x%5E3%20%2B%202%20x%5E2%20-%20x%20-2%20%3D%20x%5E2%20%28x%2B2%29%20-1%20%28x%2B2%29%20%3D%20%28x%5E2%20-1%29%20%28x%20%2B%202%29%20%3D%20%28x-1%29%20%28x%20%2B%201%29%20%28x%20%2B%202%29" \* MERGEFORMATINET  and thus has zeros at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20%5Cpm%201%2C%20-2." \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/images?id=123715" \* MERGEFORMATINET  Asymptotes and limits at infinity Given the domain, we note that there are no vertical asymptotes. We note that  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Clim_%7Bx%5Cto%20%5Cinfty%7D%20f%28x%29%3D%2B%5Cinfty" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Clim_%7Bx%5Cto%20-%5Cinfty%7D%20f%28x%29%3D-%5Cinfty." \* MERGEFORMATINET  Differentiability  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%28x%29%20%3D%203%20x%20%5E2%20%2B%204%20x%20-1%20%3D%200" \* MERGEFORMATINET if  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%3D%5Cfrac%7B-4%5Cpm%5Csqrt%7B28%7D%7D%7B6%7D%3D%5Cfrac%7B-2%5Cpm%5Csqrt%7B7%7D%7D%7B3%7D" \* MERGEFORMATINET . These are the critical values. We note that the function is differentiable at every point of its domain.  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%28x%29%20%3E%200" \* MERGEFORMATINET on  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cleft%20%28-%5Cinfty%2C%5Cfrac%7B-2-%5Csqrt%7B7%7D%7D%7B3%7D%5Cright%20%29%0A" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cleft%20%28%5Cfrac%7B-2%2B%5Csqrt%7B7%7D%7D%7B3%7D%2C%20%2B%5Cinfty%5Cright%20%29" \* MERGEFORMATINET ; hence the function is increasing in these intervals. Similarly,  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%28x%29%20%3C%200" \* MERGEFORMATINET on  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cleft%20%28%5Cfrac%7B-2-%5Csqrt%7B7%7D%7D%7B3%7D%2C%5Cfrac%7B-2%2B%5Csqrt%7B7%7D%7D%7B3%7D%5Cright%20%29" \* MERGEFORMATINET and thus is  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET decreasing there.  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%27%28x%29%20%3D%206x%20%2B%204%20%3D%200" \* MERGEFORMATINET if  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%3D-%5Cfrac%7B2%7D%7B3%7D%2C" \* MERGEFORMATINET where there is an inflection point. In addition,  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%27%5Cleft%20%28%5Cfrac%7B-2-%5Csqrt%7B7%7D%7D%7B3%7D%5Cright%20%29%3C0" \* MERGEFORMATINET . Hence the graph has a relative maximum at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%3D%5Cfrac%7B-2-%5Csqrt%7B7%7D%7D%7B3%7D" \* MERGEFORMATINET and located at the point  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%28-1.55%2C0.63%29." \* MERGEFORMATINET  We note that  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%27%28x%29%20%3C%200" \* MERGEFORMATINET for  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%3C-%5Cfrac%7B2%7D%7B3%7D" \* MERGEFORMATINET . The graph is concave down in  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cleft%20%28-%5Cinfty%2C-%5Cfrac%7B2%7D%7B3%7D%5Cright%20%29." \* MERGEFORMATINET  And we have  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%27%5Cleft%20%28%5Cfrac%7B-2%2B%5Csqrt%7B7%7D%7D%7B3%7D%5Cright%20%29%3E0" \* MERGEFORMATINET ; hence the graph has a relative minimum at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20%5Cfrac%7B-2%2B%5Csqrt%7B7%7D%7D%7B3%7D" \* MERGEFORMATINET and located at the point  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%280.22%2C%20-2.11%29." \* MERGEFORMATINET  We note that  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%27%28x%29%3E%200" \* MERGEFORMATINET for  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%3E-%5Cfrac%7B2%7D%7B3%7D." \* MERGEFORMATINET The graph is concave up in  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cleft%20%28-%5Cfrac%7B2%7D%7B3%7D%2C%20%2B%5Cinfty%5Cright%20%29." \* MERGEFORMATINET  Table Summary  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%20x%5E3%20%2B%202%20x%5E2%20-%20x%20-%202" \* MERGEFORMATINET AnalysisDomain and Range INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=D%3D%28-%5Cinfty%2C%2B%5Cinfty%29%2C%20R%3D%5Cleft%20%5C%7B%20%5Cright%20.%20%5Cmathrm%7Ball%5C%20reals%7D%5Cleft%20.%20%5Cright%20%5C%7D" \* MERGEFORMATINET Intercepts and Zeroszeros at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20%5Cpm%201%2C%20-2%2C" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=y%2C" \* MERGEFORMATINET intercept at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%280%2C-2%29" \* MERGEFORMATINET Asymptotes and limits at infinityno asymptotes Differentiabilitydifferentiable at every point of its domain Intervals where  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET is increasing INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cleft%20%28-%5Cinfty%2C%5Cfrac%7B-2-%5Csqrt%7B7%7D%7D%7B3%7D%5Cright%20%29" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cleft%20%28%5Cfrac%7B-2%2B%5Csqrt%7B7%7D%7D%7B3%7D%2C%2B%5Cinfty%5Cright%20%29" \* MERGEFORMATINET Intervals where  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET is decreasing INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cleft%20%28%5Cfrac%7B-2-%5Csqrt%7B7%7D%7D%7B3%7D%2C%5Cfrac%7B-2%2B%5Csqrt%7B7%7D%7D%7B3%7D%5Cright%20%29" \* MERGEFORMATINET Relative extremarelative maximum at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%3D%5Cfrac%7B-2-%5Csqrt%7B7%7D%7D%7B3%7D" \* MERGEFORMATINET and located at the point  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%28-1.55%2C0.63%29" \* MERGEFORMATINET ; relative minimum at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%3D%5Cfrac%7B-2%2B%5Csqrt%7B7%7D%7D%7B3%7D" \* MERGEFORMATINET and located at the point  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%280.22%2C-2.11%29." \* MERGEFORMATINET Concavityconcave up in  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cleft%20%28-%5Cfrac%7B2%7D%7B3%7D%2C%2B%5Cinfty%5Cright%20%29" \* MERGEFORMATINET . concave down in  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cleft%20%28-%5Cinfty%2C-%5Cfrac%7B2%7D%7B3%7D%5Cright%20%29" \* MERGEFORMATINET . Inflection points INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%3D-%5Cfrac%7B2%7D%7B3%7D" \* MERGEFORMATINET , located at the point  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cleft%20%28-%5Cfrac%7B2%7D%7B3%7D%2C-.74%5Cright%20%29" \* MERGEFORMATINET Here is a sketch of the graph:  INCLUDEPICTURE "http://www.ck12.org/ck12/images?id=123755" \* MERGEFORMATINET  Example 3: Analyzing Radical Functions Consider the function  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%3D%5Csqrt%7B2x-1%7D." \* MERGEFORMATINET  General Properties The domain of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET is  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cleft%20%28%5Cfrac%7B1%7D%7B2%7D%2C%2B%5Cinfty%5Cright%20%29" \* MERGEFORMATINET , and it has a zero at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%3D%5Cfrac%7B1%7D%7B2%7D." \* MERGEFORMATINET  Asymptotes and Limits at Infinity Given the domain, we note that there are no vertical asymptotes. We note that  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Clim_%7Bx%5Cto%20%5Cinfty%7D%20f%28x%29%3D%2B%5Cinfty" \* MERGEFORMATINET . Differentiability  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%28x%29%3D%5Cfrac%7B1%7D%7B%5Csqrt%7B2x-1%7D%7D%3E0" \* MERGEFORMATINET for the entire domain of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f." \* MERGEFORMATINET Hence  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET is increasing everywhere in its domain.  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%28x%29" \* MERGEFORMATINET is not defined at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%3D%5Cfrac%7B1%7D%7B2%7D" \* MERGEFORMATINET , so  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%3D%5Cfrac%7B1%7D%7B2%7D" \* MERGEFORMATINET is a critical value.  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%27%28x%29%3D%5Cfrac%7B-1%7D%7B%5Csqrt%7B%282x-1%29%5E3%7D%7D%3C0" \* MERGEFORMATINET everywhere in  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cleft%20%28%5Cfrac%7B1%7D%7B2%7D%2C%2B%5Cinfty%5Cright%20%29" \* MERGEFORMATINET . Hence  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET is concave down in  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cleft%20%28%5Cfrac%7B1%7D%7B2%7D%2C%2B%5Cinfty%5Cright%20%29." \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%28x%29" \* MERGEFORMATINET is not defined at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%3D%5Cfrac%7B1%7D%7B2%7D" \* MERGEFORMATINET , so  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%3D%5Cfrac%7B1%7D%7B2%7D" \* MERGEFORMATINET is an absolute minimum. Table Summary INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%3D%5Csqrt%7B2x-1%7D" \* MERGEFORMATINET AnalysisDomain and Range INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=D%3D%5Cleft%20%28%5Cfrac%7B1%7D%7B2%7D%2C%2B%5Cinfty%5Cright%20%29%2CR%3D%20%5Cleft%20%5C%7By%5Cge0%20%5Cright%20%5C%7D" \* MERGEFORMATINET Intercepts and Zeroszeros at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%3D%5Cfrac%7B1%7D%7B2%7D" \* MERGEFORMATINET , no  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=y-" \* MERGEFORMATINET intercept Asymptotes and limits at infinityno asymptotes Differentiabilitydifferentiable in  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cleft%20%28%5Cfrac%7B1%7D%7B2%7D%2C%2B%5Cinfty%5Cright%20%29" \* MERGEFORMATINET Intervals where  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET is increasingeverywhere in  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20D%3D%5Cleft%20%28%5Cfrac%7B1%7D%7B2%7D%2C%2B%5Cinfty%5Cright%20%29" \* MERGEFORMATINET Intervals where  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET is decreasingnowhere Relative extremanone absolute minimum at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%3D%5Cfrac%7B1%7D%7B2%7D" \* MERGEFORMATINET , located at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cleft%20%28%5Cfrac%7B1%7D%7B2%7D%2C0%5Cright%20%29" \* MERGEFORMATINET Concavityconcave down in  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cleft%20%28%5Cfrac%7B1%7D%7B2%7D%2C%2B%5Cinfty%5Cright%20%29" \* MERGEFORMATINET Inflection pointsnone Here is a sketch of the graph:  INCLUDEPICTURE "http://www.ck12.org/ck12/images?id=123759" \* MERGEFORMATINET  Example 4: Analyzing Trigonometric Functions We will see that while trigonometric functions can be analyzed using what we know about derivatives, they will provide some interesting challenges that we will need to address. Consider the function  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%20%3D%20x%20-2%20%5Csin%20x" \* MERGEFORMATINET on the interval  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5B-%5Cpi%2C%5Cpi%5D." \* MERGEFORMATINET  General Properties We note that  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET is a continuous function and thus attains an absolute maximum and minimum in  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5B-%5Cpi%2C%5Cpi%5D." \* MERGEFORMATINET Its domain is  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5B-%5Cpi%2C%5Cpi%5D" \* MERGEFORMATINET and its range is  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20R%3D%5Cleft%20%5C%7B-%5Cpi%20%5Cle%20y%20%5Cle%20%5Cpi%5Cright%20%5C%7D." \* MERGEFORMATINET  Differentiability  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%28x%29%20%3D%201%20-%202%20%5Ccos%20x%20%3D%200" \* MERGEFORMATINET at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20x%20%3D%20-%20%5Cfrac%7B%5Cpi%7D%20%20%7B3%7D%2C%5Cfrac%7B%5Cpi%7D%7B3%7D" \* MERGEFORMATINET . Note that  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%28x%29%20%3E%200" \* MERGEFORMATINET on  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cleft%20%28%5Cfrac%7B%5Cpi%7D%20%7B3%7D%2C%20%7B%5Cpi%7D%5Cright%20%29" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cleft%20%28-%5Cpi%2C%20%5Cfrac%7B%5Cpi%7D%20%7B3%7D%5Cright%20%29" \* MERGEFORMATINET ; therefore the function is increasing in  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cleft%20%28%5Cfrac%7B%5Cpi%7D%20%7B3%7D%2C%20%5Cpi%5Cright%20%29" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cleft%20%28-%5Cpi%2C%20-%5Cfrac%7B%5Cpi%7D%20%7B3%7D%5Cright%20%29" \* MERGEFORMATINET . Note that  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%28x%29%20%3C%200" \* MERGEFORMATINET on  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cleft%20%28-%5Cfrac%7B%5Cpi%7D%20%7B3%7D%2C%5Cfrac%7B%5Cpi%7D%7B3%7D%5Cright%20%29" \* MERGEFORMATINET ; therefore the function is decreasing in  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cleft%20%28-%5Cfrac%7B%5Cpi%7D%20%7B3%7D%2C%5Cfrac%7B%5Cpi%7D%7B3%7D%5Cright%20%29" \* MERGEFORMATINET .  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%27%28x%29%20%3D%202%20%5Csin%20x%20%3D%200" \* MERGEFORMATINET if  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%200%2C%20%5Cpi%2C%20-%5Cpi." \* MERGEFORMATINET Hence the critical values are at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20x%20%3D%20-%20%5Cpi%2C%20-%5Cfrac%7B%5Cpi%7D%20%7B3%7D%2C%200%2C%20%5Cfrac%7B%5Cpi%7D%20%7B3%7D%2C%20%5Cpi." \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20f%27%27%20%5Cleft%20%28%5Cfrac%7B%5Cpi%7D%20%7B3%7D%5Cright%20%29%20%3E%200%3B" \* MERGEFORMATINET hence there is a relative minimum at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20%5Cfrac%7B%5Cpi%7D%20%7B3%7D." \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20f%27%27%5Cleft%20%28-%5Cfrac%7B%5Cpi%7D%20%7B3%7D%5Cright%20%29%20%3C%200" \* MERGEFORMATINET ; hence there is a relative maximum at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20-%5Cfrac%7B%5Cpi%7D%20%7B3%7D." \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%27%28x%29%20%3E%200" \* MERGEFORMATINET on  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%280%2C%20%5Cpi%29" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%27%28x%29%20%3C%200" \* MERGEFORMATINET on  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%28-%5Cpi%2C%200%29." \* MERGEFORMATINET Hence the graph is concave up and decreasing on  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%280%2C%20%5Cpi%29" \* MERGEFORMATINET and concave down on  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%28-%5Cpi%2C%200%29." \* MERGEFORMATINET There is an inflection point at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%200%2C" \* MERGEFORMATINET located at the point  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%280%2C%200%29." \* MERGEFORMATINET  Finally, there is absolute minimum at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20-%5Cpi%2C" \* MERGEFORMATINET located at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%28-%5Cpi%2C%20-%5Cpi%29%2C" \* MERGEFORMATINET and an absolute maximum at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20%5Cpi%2C" \* MERGEFORMATINET located at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%28%5Cpi%2C%20%5Cpi%29." \* MERGEFORMATINET  Table Summary INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%20x%20-%202%5Csin%20x" \* MERGEFORMATINET AnalysisDomain and Range INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20D%20%3D%20%5B-%5Cpi%2C%20%5Cpi%5D%2C" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=R%3D%5Cleft%20%5C%7B-%5Cpi%20%5Cle%20y%20%5Cle%20%5Cpi%5Cright%20%5C%7D" \* MERGEFORMATINET Intercepts and Zeros INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20-%20%5Cfrac%7B%5Cpi%7D%20%7B3%7D%2C%20%5Cfrac%7B%5Cpi%7D%20%7B3%7D" \* MERGEFORMATINET Asymptotes and limits at infinityno asymptotes Differentiabilitydifferentiable in  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=D%20%3D%20%5B-%5Cpi%2C%20%5Cpi%5D" \* MERGEFORMATINET Intervals where  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET is increasing INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cleft%20%28%5Cfrac%7B%5Cpi%7D%20%7B3%7D%2C%20%5Cpi%5Cright%20%29" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%28-%5Cpi%2C%20-%5Cfrac%7B%5Cpi%7D%20%7B3%7D%29" \* MERGEFORMATINET Intervals where  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET is decreasing INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cleft%20%28-%5Cfrac%7B%5Cpi%7D%20%7B3%7D%2C%20%5Cfrac%7B%5Cpi%7D%20%7B3%7D%5Cright%20%29" \* MERGEFORMATINET Relative extremarelative maximum at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20-%5Cpi/3" \* MERGEFORMATINET  relative minimum at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20%5Cpi/3" \* MERGEFORMATINET  absolute maximum at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20%5Cpi" \* MERGEFORMATINET  absolute minimum at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20%5Cfrac%7B1%7D%20%7B2%7D" \* MERGEFORMATINET , located at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cleft%20%28%5Cfrac%7B1%7D%20%7B2%7D%2C%200%5Cright%20%29" \* MERGEFORMATINET Concavityconcave up in  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%280%2C%20%5Cpi%29" \* MERGEFORMATINET Inflection points INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%200%2C" \* MERGEFORMATINET located at the point  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%280%2C%200%29" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/images?id=123758" \* MERGEFORMATINET  Lesson Summary We summarized the properties of function, including intercepts, domain, range, continuity, asymptotes, relative extreme, concavity, points of inflection, and limits at infinity. We applied the First and Second Derivative Tests to sketch graphs. Multimedia Links Each of the problems above started with a function and then we analyzed its zeros, derivative, and concavity. Even without the function definition it is possible to sketch the graph if you know some key pieces of information. In the following video the narrator illustrates how to use information about the derivative of a function and given points on the function graph to sketch the function.  HYPERLINK "http://www.youtube.com/watch?v=ojcp0GJKluM" Khan Academy Graphing with Calculus (9:43) INCLUDEPICTURE "http://i4.ytimg.com/vi/ojcp0GJKluM/hqdefault.jpg" \* MERGEFORMATINET . Another approach to this analysis is to look at a function, its derivative, and its second derivative on the same set of axes. This interactive applet called  HYPERLINK "http://calculusapplets.com/curveanalysisbasic.html" Curve Analysis allows you to trace function points on a graph and its first and second derivative. You can also enter new functions (including the ones from the examples above) to analyze the functions and their derivatives. For more information about computing derivatives of higher orders (7.0), see  HYPERLINK "http://www.youtube.com/watch?v=_-_VkzdNok4" Math Video Tutorials by James Sousa, Higher-Order Derivatives: Part 1 of 2 (7:34) INCLUDEPICTURE "http://i4.ytimg.com/vi/_-_VkzdNok4/hqdefault.jpg" \* MERGEFORMATINET  and  HYPERLINK "http://www.youtube.com/watch?v=SdQXLxr0P0A" Math Video Tutorials by James Sousa, Higher-Order Derivatives: Part 2 of 2 (5:21) INCLUDEPICTURE "http://i4.ytimg.com/vi/SdQXLxr0P0A/hqdefault.jpg" \* MERGEFORMATINET . For a presentation of higher order partial derivatives (7.0), see  HYPERLINK "http://www.youtube.com/watch?v=AWrXr9ozeOU" Calculus, Higher Order Derivatives (8:09)  INCLUDEPICTURE "http://i2.ytimg.com/vi/AWrXr9ozeOU/hqdefault.jpg" \* MERGEFORMATINET . Review Questions Summarize each of the following functions by filling out the table. Use the information to sketch a graph of the function. AnalysisDomain and RangeIntercepts and ZerosAsymptotes and limits at infinityDifferentiabilityIntervals where  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET is increasingIntervals where  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET is decreasingRelative extremaConcavityInflection points INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%20x%20%5E3%20%2B%203%20x%5E2%20-%20x%20-%203" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%20-x%5E4%20%2B%204x%5E3%20-%204x%5E2" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%20%5Cfrac%7B2x%20-2%7D%20%7Bx%5E2%7D" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%20x%20-%20x%5E%5Cfrac%7B1%7D%20%7B3%7D" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%20-%5Csqrt%7B2x%20-%206%7D%20%2B3" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%20x%5E2%20-2%5Csqrt%7Bx%7D" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%28x%29%20%3D%201%20%2B%20%5Ccos%20x%20%3D%200" \* MERGEFORMATINET  on [-(, (] Review Answers 1.  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%20x%5E3%20%2B%203x%5E2%20-%20x%20-%203" \* MERGEFORMATINET AnalysisDomain and Range INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20D%20%3D%20%28-%5Cinfty%2C%20%2B%5Cinfty%29%2C" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20R%20%3D%5Cleft%20%5C%7B%5Cmathrm%7Ball%5C%20reals%7D%5Cright%20%5C%7D" \* MERGEFORMATINET Intercepts and Zeroszeros at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20%5Cpm%201%2C%20-3%2C" \* MERGEFORMATINET  y-intercept at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%280%2C%20-3%29" \* MERGEFORMATINET Asymptotes and limits at infinityno asymptotes Differentiabilitydifferentiable at every point of its domain Intervals where  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET is increasing INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cleft%20%28-%5Cinfty%2C%20%5Cfrac%7B-3-2%5Csqrt%7B3%7D%7D%20%7B3%7D%5Cright%20%29" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cleft%20%28%5Cfrac%7B-3%20%2B%202%5Csqrt%7B3%7D%7D%20%7B3%7D%20%2C%20%2B%5Cinfty%5Cright%20%29" \* MERGEFORMATINET Intervals where  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET is decreasing INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cleft%28%5Cfrac%7B-3-2%5Csqrt%7B3%7D%7D%20%7B3%7D%20%2C%20%5Cfrac%7B-3%20%2B%202%5Csqrt%7B3%7D%7D%20%7B3%7D%5Cright%29" \* MERGEFORMATINET Relative extremarelative maximum at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20%5Cfrac%7B-3-2%5Csqrt%7B3%7D%7D%20%7B3%7D%2C" \* MERGEFORMATINET  located at the point  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%28-2.15%2C%203.07%29" \* MERGEFORMATINET ; relative minimum at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20x%20%3D%20%5Cfrac%7B-3%2B2%5Csqrt%7B3%7D%7D%20%7B3%7D%2C" \* MERGEFORMATINET  located at the point  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%280.15%2C%20-3.07%29" \* MERGEFORMATINET Concavityconcave up in  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%28-1%2C%20%2B%5Cinfty%29" \* MERGEFORMATINET  concave down in  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%28-%5Cinfty%2C%20-1%29" \* MERGEFORMATINET Inflection points INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20-1%2C" \* MERGEFORMATINET  located at the point  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%28-1%2C%200%29" \* MERGEFORMATINET 2.  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%20-x%5E4%20%2B%204x%5E3%20-%204x%5E2" \* MERGEFORMATINET AnalysisDomain and Range INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20D%20%3D%20%28-%5Cinfty%2C%20%2B%5Cinfty%29%2C" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=R%20%3D%20%5Cleft%20%5C%7By%20%5Cle%200%5Cright%20%5C%7D" \* MERGEFORMATINET Intercepts and Zeroszeros at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%200%2C%202%2C" \* MERGEFORMATINET  y-intercept at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%280%2C%200%29" \* MERGEFORMATINET Asymptotes and limits at infinityno asymptotes Differentiabilitydifferentiable at every point of its domain Intervals where  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET is increasing INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%28-%5Cinfty%2C%200%29" \* MERGEFORMATINET  and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%281%2C%202%29" \* MERGEFORMATINET Intervals where  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET is decreasing INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%280%2C%201%29" \* MERGEFORMATINET  and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%282%2C%20%2B%5Cinfty%29" \* MERGEFORMATINET Relative extremarelative maximum at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%200" \* MERGEFORMATINET , located at the point  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%280%2C%200%29" \* MERGEFORMATINET ; and at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%202" \* MERGEFORMATINET  located at the point  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%282%2C%200%29" \* MERGEFORMATINET  relative minimum at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%201" \* MERGEFORMATINET , located at the point  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%281%2C%20-1%29" \* MERGEFORMATINET Concavityconcave up in  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cleft%20%28%5Cfrac%7B2%7D%20%7B3%7D%2C%20%5Cfrac%7B5%7D%20%7B3%7D%5Cright%20%29" \* MERGEFORMATINET  concave down in  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cleft%20%28-%5Cinfty%2C%20%5Cfrac%7B2%7D%20%7B3%7D%5Cright%20%29" \* MERGEFORMATINET  and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cleft%20%28%5Cfrac%7B5%7D%20%7B3%7D%2C%20%2B%5Cinfty%5Cright%20%29" \* MERGEFORMATINET Inflection points INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%3D%20%5Cfrac%7B2%7D%20%7B3%7D%2C%20%5Cfrac%20%7B5%7D%20%7B3%7D" \* MERGEFORMATINET , located at the points  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cleft%20%28%5Cfrac%7B2%7D%20%7B3%7D%2C%20-0.79%5Cright%20%29" \* MERGEFORMATINET  and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cleft%20%28%5Cfrac%7B5%7D%20%7B3%7D%2C%20-0.30%5Cright%20%29" \* MERGEFORMATINET 3.  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20f%28x%29%20%3D%20%5Cfrac%7B2x%20-%202%7D%20%7Bx%5E2%7D" \* MERGEFORMATINET AnalysisDomain and Range INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20D%20%3D%20%28-%5Cinfty%2C%200%29%20%5Ccup%20%280%2C%20%2B%5Cinfty%29%2C" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=R%20%3D%20%5Cleft%20%5C%7By%20%5Cneq%200%20%5Cright%20%5C%7D" \* MERGEFORMATINET Intercepts and Zeroszeros at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%201" \* MERGEFORMATINET , no y-intercept Asymptotes and limits at infinityHA  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=y%20%3D%200" \* MERGEFORMATINET Differentiabilitydifferentiable at every point of its domain Intervals where  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET is increasing INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%280%2C%202%29" \* MERGEFORMATINET Intervals where  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET is decreasing INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%28-%5Cinfty%2C%200%29" \* MERGEFORMATINET  and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%282%2C%20%2B%5Cinfty%29" \* MERGEFORMATINET Relative extremarelative maximum at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%202%2C" \* MERGEFORMATINET  located at the point  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%282%2C%200.5%29" \* MERGEFORMATINET Concavityconcave up in  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%283%2C%20%2B%5Cinfty%29" \* MERGEFORMATINET  concave down in  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%28-%5Cinfty%2C%200%29" \* MERGEFORMATINET  and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%280%2C%203%29" \* MERGEFORMATINET Inflection points INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%203%2C" \* MERGEFORMATINET  located at the point  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%283%2C%20%5Cfrac%7B4%7D%20%7B9%7D%29" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20f%28x%29%20%3D%20x%20-%20x%5E%5Cfrac%7B1%7D%20%7B3%7D" \* MERGEFORMATINET AnalysisDomain and Range INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=D%20%3D%20%28-%5Cinfty%2C%20%2B%5Cinfty%29%2C" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=R%20%3D%20%5Cleft%20%5C%7B%5Cmathrm%7Ball%5C%20reals%7D%20%5Cright%20%5C%7D" \* MERGEFORMATINET Intercepts and Zeroszeros at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20%5Cpm%201%2C%200%2C" \* MERGEFORMATINET  y-intercept at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%280%2C%200%29" \* MERGEFORMATINET Asymptotes and limits at infinityno asymptotes Differentiabilitydifferentiable in  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%28%5Cinfty%2C%200%29%20%5Ccup%20%280%2C%20%2B%5Cinfty%29" \* MERGEFORMATINET Intervals where  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET is increasing INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cleft%20%28-%5Cinfty%2C%20%5Cfrac%7B-%5Csqrt%7B3%7D%7D%20%7B9%7D%5Cright%20%29" \* MERGEFORMATINET  and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cleft%20%28%5Cfrac%7B%5Csqrt%7B3%7D%7D%20%7B9%7D%2C%20%2B%5Cinfty%5Cright%20%29" \* MERGEFORMATINET Intervals where  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET is decreasing INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cleft%20%28%5Cfrac%7B-%5Csqrt%7B3%7D%7D%20%7B9%7D%2C%20%5Cfrac%7B%5Csqrt%7B3%7D%7D%20%7B9%7D%5Cright%20%29" \* MERGEFORMATINET Relative extremarelative maximum at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cmathit%20x%20%3D%20%5Cfrac%7B-%5Csqrt%7B3%7D%7D%20%7B9%7D%2C" \* MERGEFORMATINET  located at the point  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cleft%20%28%5Cfrac%7B-%5Csqrt%7B3%7D%7D%20%7B9%7D%2C%200.384%5Cright%20%29" \* MERGEFORMATINET  relative minimum at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20x%20%3D%20%5Cfrac%7B%5Csqrt%7B3%7D%7D%20%7B9%7D%2C" \* MERGEFORMATINET  located at the point  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cleft%20%28%5Cfrac%7B%5Csqrt%7B3%7D%7D%20%7B9%7D%2C%20-%200.384%5Cright%20%29" \* MERGEFORMATINET Concavityconcave up in  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%280%2C%20%2B%5Cinfty%29" \* MERGEFORMATINET  concave down in  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%28-%5Cinfty%2C%200%29" \* MERGEFORMATINET Inflection points INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%200%2C" \* MERGEFORMATINET  located at the point  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%280%2C%200%29" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20f%28x%29%20%3D%20-%20%5Csqrt%7B2x%20-%206%7D%20%2B%203" \* MERGEFORMATINET AnalysisDomain and Range INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20D%20%3D%20%283%2C%20%2B%5Cinfty%29%2C" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=R%20%3D%20%5Cleft%20%5C%7By%20%5Cle%203%20%5Cright%20%5C%7D" \* MERGEFORMATINET Intercepts and Zeroszero at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20x%20%3D%20%5Cfrac%7B15%7D%20%7B2%7D%2C" \* MERGEFORMATINET  no y-intercept Asymptotes and limits at infinityno asymptotes Differentiabilitydifferentiable in  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%283%2C%20%2B%5Cinfty%29" \* MERGEFORMATINET Intervals where  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET is increasingnowhere Intervals where  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET is decreasingeverywhere in  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20D%20%3D%20%283%2C%20%2B%20%5Cinfty%29" \* MERGEFORMATINET Relative extremanone absolute maximum at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%203%2C" \* MERGEFORMATINET  located at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%283%2C%203%29" \* MERGEFORMATINET Concavityconcave up in  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%283%2C%20%2B%5Cinfty%29" \* MERGEFORMATINET Inflection pointsnone   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20f%28x%29%20%3D%20x%5E2%20-%202%5Csqrt%7Bx%7D" \* MERGEFORMATINET AnalysisDomain and Range INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=D%20%3D%20%280%2C%20%2B%5Cinfty%29%2C" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=R%20%3D%20%5Cleft%20%5C%7By%20%5Cge%20-%201.19%20%5Cright%20%5C%7D" \* MERGEFORMATINET Intercepts and Zeroszero at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20%5Csqrt%5B3%5D%7B4%7D%2C" \* MERGEFORMATINET  no y-intercept Asymptotes and limits at infinityno asymptotes Differentiabilitydifferentiable in  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%280%2C%20%2B%5Cinfty%29" \* MERGEFORMATINET Intervals where  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET is increasing INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cleft%20%28%5Cfrac%7B%5Csqrt%5B3%5D%7B16%7D%7D%20%7B4%7D%2C%20%2B%5Cinfty%5Cright%20%29" \* MERGEFORMATINET Intervals where  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET is decreasing INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cleft%20%280%2C%20%5Cfrac%7B%5Csqrt%5B3%5D%7B16%7D%7D%20%7B4%7D%5Cright%20%29" \* MERGEFORMATINET Relative extremarelative minimum at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20x%20%3D%20%5Cfrac%7B%5Csqrt%5B3%5D%7B16%7D%7D%20%7B4%7D%2C" \* MERGEFORMATINET  located at the point  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20x%20%3D%20%5Cleft%20%28%5Cfrac%7B%5Csqrt%5B3%5D%7B16%7D%7D%20%7B4%7D%2C%20-1.19%5Cright%20%29" \* MERGEFORMATINET Concavityconcave up in  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%280%2C%20%2B%5Cinfty%29" \* MERGEFORMATINET Inflection pointsnone 7.  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%28x%29%20%3D%201%20%2B%20%5Ccos%20x" \* MERGEFORMATINET AnalysisDomain and Range INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20D%20%3D%20%5B-%20%5Cpi%2C%20%5Cpi%5D%2C" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=R%20%3D%20%5Cleft%20%5C%7B0%20%5Cle%20y%20%5Cle%202%20%5Cright%20%5C%7D" \* MERGEFORMATINET Intercepts and Zeroszeros at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20-%20%5Cpi%2C%20%5Cpi%2C" \* MERGEFORMATINET , y-intercept at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%280%2C%202%29" \* MERGEFORMATINET Asymptotes and limits at infinityno asymptotes;  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Clim_%7Bx%20%5Cleftrightarrow%20%5Cinfty%7D%20f%28x%29" \* MERGEFORMATINET  does not exist Differentiabilitydifferentiable at every point of its domain Intervals where  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET is increasing INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%28-%20%5Cpi%2C%200%29" \* MERGEFORMATINET Intervals where  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET is decreasing INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%280%2C%20%5Cpi%29" \* MERGEFORMATINET Relative extremaabsolute max at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%200%2C" \* MERGEFORMATINET  located at the point  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%280%2C%202%29" \* MERGEFORMATINET  absolute minimums at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20%5Cpm%20%5Cpi%2C" \* MERGEFORMATINET  located at the points  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%28-%5Cpi%2C%200%29" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%28%5Cpi%2C%200%29" \* MERGEFORMATINET Concavityconcave down in  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%28-%5Cpi%2C%20%5Cpi%29" \* MERGEFORMATINET Inflection points INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20x%20%3D%20%5Cpm%20%5Cfrac%7B%5Cpi%7D%20%7B2%7D" \* MERGEFORMATINET , located at the points  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cleft%20%28-%5Cfrac%7B%5Cpi%7D%20%7B2%7D%2C%201%5Cright%20%29" \* MERGEFORMATINET  and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cleft%20%28%5Cfrac%7B%5Cpi%7D%20%7B2%7D%2C%201%5Cright%20%29" \* MERGEFORMATINET  Curve Sketching HW 1 For each function, find all requested info, and sketch the graph. Use your calculator to check your sketch. 1.)  EMBED Equation.DSMT4  INC: DEC: MAX: MIN: C. UP: C. DOWN: IP: Roots: Holes: VA:  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  y-int: Other point (if needed):  2.)  EMBED Equation.DSMT4  INC: DEC: MAX: MIN: C. UP: C. DOWN: IP: Roots: Holes: VA:  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  y-int: Other point (if needed):  3.)  EMBED Equation.DSMT4  INC: DEC: MAX: MIN: C. UP: C. DOWN: IP: Roots: Holes: VA:  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  y-int: Other point (if needed):  Hints: 1.)  EMBED Equation.DSMT4  2.)  EMBED Equation.DSMT4  3.)  EMBED Equation.DSMT4  Curve Sketching HW 2 For each function, find all requested info, and sketch the graph. Use your calculator to check your sketch. 1.)  EMBED Equation.DSMT4  INC: DEC: MAX: MIN: C. UP: C. DOWN: IP: Roots: Holes: VA:  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  y-int: Other point (if needed):  2.)  EMBED Equation.DSMT4  INC: DEC: MAX: MIN: C. UP: C. DOWN: IP: Roots: Holes: VA:  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  y-int: Other point (if needed):  3.)  EMBED Equation.DSMT4  INC: DEC: MAX: MIN: C. UP: C. DOWN: IP: Roots: Holes: VA:  EMBED Equation.DSMT4  ';[ \     5 6 ҺqWA+h?h:95B*CJOJQJ\^Jph2jh?h:9B*CJOJQJU^JaJph2jh?h:9B*CJOJQJU^JaJph2jh?h:9B*CJOJQJU^JaJph)h?h:9B*CJOJQJ^JaJph/h?h:95B*CJOJQJ\^JaJph/h?h:95B*CJOJQJ\^JaJph)hP55B*CJOJQJ\^JaJph';W} $  * 2odgd:9 *d^*gd:9 & F*ddd[$\$^*gd:9ddd[$\$gd:9ddd@&[$\$gd:9dYdd@&[$\$gd:9 T U V W _ `   c d e f l жМЂhN2j?h?h:9B*CJOJQJU^JaJph2j~ h?h:9B*CJOJQJU^JaJph2j h?h:9B*CJOJQJU^JaJph2jh?h:9B*CJOJQJU^JaJph2jh?h:9B*CJOJQJU^JaJph2jh?h:9B*CJOJQJU^JaJph)h?h:9B*CJOJQJ^JaJphl s w x 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