ࡱ> supqrfIVIVV [@ bjbj44 oViVi MMMMYYY8Y^M:aP{{{ ~Xl8%$R7r9M5{ ~*=2   b+{2M ~% % r }:><M/ ~a 0"`Yӯ  1 o<% /MMM= MMĨnDcjMMn EMBED Equation.DSMT4  EMBED Equation.DSMT4  Chapter 5 Derivative Rules The derivative of a function at x is defined as EMBED Equation.DSMT4 , which can be used to find slopes of tangent lines as well as instantaneous rates of change. Unfortunately, computing the derivative directly from the definition can be quite tedious and overwhelming. In this chapter, we will present several rules of differentiation that will greatly simplify the differentiation. 5.1 Basic Differentiation Rules Before we start to study the rules of differentiation, we need to introduce alternate ways to represent the derivative. ( Table 5.1 Alternative Notations for the Derivative For  EMBED Equation.DSMT4 , all of the following may be used to represent the derivative:  EMBED Equation.DSMT4  EMBED Equation.DSMT4  The process of finding the derivative of a function is called differentiation. A function is differentiable at x if its derivative exists at x. The following table lists the basic differentiation rules. ( Table 5.1 Basic Differentiation Rules Let C, a, and n be real numbers with EMBED Equation.DSMT4 . Let f(x) and g(x) be differentiable functions. RulesExamples1. Constant rule: If  EMBED Equation.DSMT4 , then  EMBED Equation.DSMT4 . If  EMBED Equation.DSMT4 , then  EMBED Equation.DSMT4 .2. Power rule: If  EMBED Equation.DSMT4 , then  EMBED Equation.DSMT4 . If  EMBED Equation.DSMT4 , then  EMBED Equation.DSMT4 . 3. If  EMBED Equation.DSMT4 , then  EMBED Equation.DSMT4 .If  EMBED Equation.DSMT4 , then  EMBED Equation.DSMT4 .4. Sum and Difference rule: If  EMBED Equation.DSMT4 , then  EMBED Equation.DSMT4 If  EMBED Equation.DSMT4 , then  EMBED Equation.DSMT4 .5. If  EMBED Equation.DSMT4 , then  EMBED Equation.DSMT4 .If  EMBED Equation.DSMT4 , then  EMBED Equation.DSMT4 . 6. If  EMBED Equation.DSMT4 , then  EMBED Equation.DSMT4 .If  EMBED Equation.DSMT4 , then  EMBED Equation.DSMT4 7. If  EMBED Equation.DSMT4 , then  EMBED Equation.DSMT4 . If  EMBED Equation.DSMT4 , then  EMBED Equation.DSMT4 .8. If  EMBED Equation.DSMT4 , then  EMBED Equation.DSMT4 . If  EMBED Equation.DSMT4 , then  EMBED Equation.DSMT4 .  ( Example 5.1 Find the derivatives: (a)  EMBED Equation.DSMT4  (b)  EMBED Equation.DSMT4  (c)  EMBED Equation.DSMT4  (d)  EMBED Equation.DSMT4  (e)  EMBED Equation.DSMT4  ( Solutions (a)   EMBED Equation.DSMT4  (b) If a power function involves negative powers, we must rewrite the function in the form of EMBED Equation.DSMT4 . Then, we apply the power rule. Recall that  EMBED Equation.DSMT4 . Hence,  (c) If a power function involves radicals, we must rewrite each radical as a fractional power. Again, we then apply the power rule. Recall that  EMBED Equation.DSMT4 . Hence,  EMBED Equation.DSMT4    EMBED Equation.DSMT4   EMBED Equation.DSMT4  (d) Using rules 5 and 6, we have   EMBED Equation.DSMT4 . (e) By rules 7 and 8, we have  EMBED Equation.DSMT4  ( Applications Since the slope of a line tangent to a curve is given by the derivative, differentiation rules can be used to find the equation of the tangent line. The steps for deriving a tangent line from a function f(x) at x = a are summarized in the following table. ( Table 5.2 How to Find a Tangent Line Assuming that f(x) is differentiable at x = a. Step 1. Find the y coordinate of the point at x = a:  EMBED Equation.DSMT4  (x and y are both given sometimes.)Step 2. Find the slope of the tangent line at x = a:  EMBED Equation.DSMT4 Step 3. Use the point-slope formula EMBED Equation.DSMT4 to find the equation of the tangent line:  EMBED Equation.DSMT4  ( Example 5.2 Find the equation of the line tangent to the graph of  EMBED Equation.DSMT4  at x = 1. Sketch the graph of f(x) and the tangent line on the same axes. ( Solutions When x = 1, we can get the y coordinate by finding f(1):  EMBED Equation.DSMT4 . By taking the derivative, we can find a formula for the slope of a line tangent to any point of f :  EMBED Equation.DSMT4  In particular, the slope of the tangent line of f at x = 1 is  EMBED Equation.DSMT4  By using the point-slope formula of the line equation with slope 2 and the point (1, 1), we have  EMBED Equation.DSMT4  The graph of  EMBED Equation.DSMT4  and the tangent line  EMBED Equation.DSMT4 at x = 1 can be seen in Figure 5.1. ( Figure 5.1 Graph of Example 5.2  ( The derivative of a function can also be interpreted as the instantaneous rate of change of f(x). Hence, we can use the derivative to study rates of change. ( Example 5.3 The population of a fire ant colony is growing according to the function  EMBED Equation.DSMT4 , where t is time measured in days. What is the rate of change of the population with respect to time when t = 4? Give units and interpret the answer. ( Solutions The rate of change of the population with respect to time is given by  EMBED Equation.DSMT4 . Therefore,   EMBED Equation.DSMT4  When t = 4, the rate is  EMBED Equation.DSMT4  If  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 also can be written as  EMBED Equation.DSMT4 in what is called Leibnizs notation. Leibnizs notation is useful for determining the units of the derivative:  EMBED Equation.DSMT4  Since N is measured in number of the fire ants and t is measured in days,  EMBED Equation.DSMT4  must be measured in number of fire ants per day. The statement  EMBED Equation.DSMT4  means that on the fourth day, the colony is increasing in size at the rate of 200 fire ants per day. This is an instantaneous rate of change, meaning that if the rate were remaining 200 fire ants increasing per day for a whole day on forth day, there would be extra 200 fire ants on the fifth day. ( ( Example 5.4 Derek purchased a new car for $20,000 dollars and its value after t years is estimated by  EMBED Equation.DSMT4 . How much was the car decreasing in value per year when it was driven out of the show room? ( Solutions This is asking for the rate of change of the cars value at t = 0, or  EMBED Equation.DSMT4 . Since  EMBED Equation.DSMT4 , we have  EMBED Equation.DSMT4  The statement  EMBED Equation.DSMT4 means that the value of the car decreased at a rate of $2,107.2 per year when the car was driven out of the show room. ( 5.2 The Product and Quotient Rule If f(x) and g(x) are both differentiable, then the derivative of  EMBED Equation.DSMT4 can be found by the product rule. ( Table 5.2 The Product Rule If EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 exist, and  EMBED Equation.DSMT4 , then   EMBED Equation.DSMT4  ( Example 5.5 Find the derivative of the following functions. (a)  EMBED Equation.DSMT4  (b)  EMBED Equation.DSMT4  ( Solutions (a)  EMBED Equation.DSMT4  can be thought of as EMBED Equation.DSMT4 where  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 . Then  EMBED Equation.DSMT4  Note: In general, the derivative of the product of two functions is not given by the product of the derivatives of the functions:  EMBED Equation.DSMT4  In this case,  EMBED Equation.DSMT4 . (b) EMBED Equation.DSMT4 can be thought as  EMBED Equation.DSMT4 where  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 . Then  EMBED Equation.DSMT4  ( If f(x) and g(x) are both differentiable, then the derivative of  EMBED Equation.DSMT4 can be found by the quotient rule. Note that we must avoid points where g(x) = 0. ( Table 5.2 The Quotient Rule If  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 exist, and  EMBED Equation.DSMT4 , then   EMBED Equation.DSMT4  which is valid for any x at which EMBED Equation.DSMT4 .  The denominator of the Quotient Rule is easy to memorize, but the numerator is not. To remember this expression, simply memorize the expression in this form:  EMBED Equation.DSMT4  ( Example 5.6 Find the derivative of the following functions. (a)  EMBED Equation.DSMT4  (b)  EMBED Equation.DSMT4  ( Solutions (a) EMBED Equation.DSMT4 can be thought as  EMBED Equation.DSMT4 , where  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 . Then   EMBED Equation.DSMT4  Note: In general, the derivative of the quotient of two functions is not given by the quotient of the derivatives of the functions:  EMBED Equation.DSMT4  In this case,  EMBED Equation.DSMT4 . (b) In this quotient,  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 . By the quotient rule, we have   EMBED Equation.DSMT4  ( Applications In economics, the word marginal refers to an instantaneous rate of change. In other words, marginal refers to the derivative. For example, if C(x) is the cost function,  EMBED Equation.DSMT4  is called the marginal cost function. The marginal cost reflects how the cost changes in response to a one unit change in the production. Knowing this cost is very important to management in their decision making processes. Similarly, if  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  are revenue and profit functions,  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 are called the marginal revenue and marginal profit functions. ( Example 5.7 The cost to produce q items is  EMBED Equation.DSMT4 dollars. Find the marginal cost of producing the 25th item. Interpret the answer in terms of costs. ( Solutions The marginal cost is defined as EMBED Equation.DSMT4 . In this case,  EMBED Equation.DSMT4  Then, the marginal cost function is   EMBED Equation.DSMT4  Hence, the marginal cost of producing the 25th item is  EMBED Equation.DSMT4  The statement  EMBED Equation.DSMT4 says that when 25 items have already made, the cost of making the next item is approximately $212.5. Another way of saying this is that it costs about $212.5 to make the 26th item. We can also find  EMBED Equation.DSMT4 by first using the distributive property before differentiation:  EMBED Equation.DSMT4  Then,  EMBED Equation.DSMT4  ( ( Example 5.8 The sales (in millions of dollars) of a DVD player t years after it is put on the market is given by  EMBED Equation.DSMT4  How fast are the sales changing at the end of the first year? At the end of the second year? ( Solutions The problems are asking for the rate at which sales are changing at the end of the first year and at the end of second year. The rate at which sales are changing at time t is given by EMBED Equation.DSMT4 . Using the quotient rule, we have  EMBED Equation.DSMT4  Therefore  EMBED Equation.DSMT4   EMBED Equation.DSMT4  Thus, sales are increasing at a rate of $7.233 million per year at the end of the first year, and sales are decreasing at a rate of $9.078 million per year at the end of the second year. ( 5.3 The Chain Rule The derivative rules we discussed in the preceding sections can let us differentiate functions like  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 . However, we still lack the tools to find the derivative of functions such as EMBED Equation.DSMT4 . Functions like  EMBED Equation.DSMT4  are called composite functions. Before we study a new rule that will let us find the derivative of a composite function, we review the composite function first. The composite functions Recall that two function can be combined in various ways (sum, difference, product and quotient) to create new functions. The composition is a special way to form a new function. ( Table 5.3 Definition of Composite function Given two function f and g, the composite function  EMBED Equation.DSMT4  is defined by  EMBED Equation.DSMT4  ( Example 5.9 If  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 , find the composite functions  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 . ( Solutions To get  EMBED Equation.DSMT4 , replace x from  EMBED Equation.DSMT4  by  EMBED Equation.DSMT4 :  EMBED Equation.DSMT4  To get  EMBED Equation.DSMT4 , replace x from  EMBED Equation.DSMT4  by  EMBED Equation.DSMT4 :  EMBED Equation.DSMT4  To get  EMBED Equation.DSMT4 , replace x from  EMBED Equation.DSMT4  by  EMBED Equation.DSMT4  itself:  EMBED Equation.DSMT4  Note that  EMBED Equation.DSMT4 , and  EMBED Equation.DSMT4  ( For a composite function EMBED Equation.DSMT4 , we might think of f(x) as the outer function that acts on the values of the inner function g(x). This point of view will help us to recognize a composite function. ( Example 5.10 Find the inner and outer functions for the following functions so that each one can be represented as a composite function. This process is called the Decomposition. (a)  EMBED Equation.DSMT4  (b)  EMBED Equation.DSMT4  (c)  EMBED Equation.DSMT4  (d)  EMBED Equation.DSMT4  ( Solutions Decomposition of Composite Functions  EMBED Equation.DSMT4 Inner Function: g(x)Outer Function: f(x)(a)  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4 (b)  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4 (c)  EMBED Equation.DSMT4  EMBED Equation.DSMT4   EMBED Equation.DSMT4 (d)  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  ( The Chain Rule In the above example, we observe that the outer and inner functions are much simpler functions that can be differentiated by the derivative rules from the previous sections. The following rule describes the relationship between the derivative of a composite function EMBED Equation.DSMT4  and derivatives of its inner and outer functions. ( Table 5.3 The Chain Rule If  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 exist, and EMBED Equation.DSMT4 , then  EMBED Equation.DSMT4 The alternative form of the Chain Rule: Let  EMBED Equation.DSMT4 , then  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 . Therefore,   EMBED Equation.DSMT4  EMBED Equation.DSMT4  When applying the Chain Rule, it is helpful to think of the composite function  EMBED Equation.DSMT4  as having two parts: an inner part and an outer part. Then, the derivative of  EMBED Equation.DSMT4 is the product of the derivative of the outer function (at the inner function u = g(x) ) and the derivative of the inner function at x.  ( Example 5.11 Find the derivative of  EMBED Equation.DSMT4 . ( Solution For this function, you can consider the inner function to be EMBED Equation.DSMT4 . Consequently, the outer function will be EMBED Equation.DSMT4 . Note   EMBED Equation.DSMT4  By the Chain Rule, we have   EMBED Equation.DSMT4     Let  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 , then  EMBED Equation.DSMT4  By the alternative form of the Chain Rule, we have   EMBED Equation.DSMT4  ( The above example is one of the most common composite functions, EMBED Equation.DSMT4 , where the outer function is a power function, EMBED Equation.DSMT4 . The rule for differentiating such function is called the General Power Rule and it can be considered a special case of the Chain Rule. ( Table 5.4 The General Power Rule If  EMBED Equation.DSMT4 exists, and EMBED Equation.DSMT4 , then  EMBED Equation.DSMT4  ( Example 5.12 Find the derivatives: (a)  EMBED Equation.DSMT4  (b)  EMBED Equation.DSMT4  (c)  EMBED Equation.DSMT4  ( Solution (a) In this function, the outer function is  EMBED Equation.DSMT4  and the inner function is  EMBED Equation.DSMT4 . By the General Power Rule, we have   EMBED Equation.DSMT4   EMBED Equation.DSMT4  (b)   EMBED Equation.DSMT4  We could also use the Quotient Rule to differentiate this function,  EMBED Equation.DSMT4  but it is much easier to use the Chain Rule. In general, do not use the Quotient Rule for functions of the form  EMBED Equation.DSMT4 . Instead, rewrite the function as  EMBED Equation.DSMT4  and use the Chain Rule instead. (c) For this function, we need to apply the Product Rule before applying the Chain Rule.   EMBED Equation.DSMT4  ( Recall that the derivative of  EMBED Equation.DSMT4 is simply EMBED Equation.DSMT4 . Can we apply the Chain Rule to functions of the form EMBED Equation.DSMT4 ? Using the Chain Rule, we find the following formulas: ( Table 5.5 The Chain For Exponential functions If  EMBED Equation.DSMT4 exists, and EMBED Equation.DSMT4 , then  EMBED Equation.DSMT4  If  EMBED Equation.DSMT4 exists, and EMBED Equation.DSMT4  with  EMBED Equation.DSMT4 , then  EMBED Equation.DSMT4  Note that the derivative of  EMBED Equation.DSMT4  is a special case of the derivative of  EMBED Equation.DSMT4 , because  EMBED Equation.DSMT4 . ( Example 5.13 Find the derivatives: (a)  EMBED Equation.DSMT4  (b)  EMBED Equation.DSMT4  (c)  EMBED Equation.DSMT4  ( Solution (a) In this function, EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 . By the Chain Rule, we have   EMBED Equation.DSMT4  (b) Let  EMBED Equation.DSMT4 , then  EMBED Equation.DSMT4 . We have  EMBED Equation.DSMT4  (c) Rewrite the function as EMBED Equation.DSMT4 .   EMBED Equation.DSMT4  This function can also be differentiated by the Product Rule.   EMBED Equation.DSMT4   EMBED Equation.DSMT4  The Product Rule is easier to remember. Hence, whenever a function can be written as a product it is more convenient to use the Product Rule than the Quotient Rule. ( Recall that the derivative of  EMBED Equation.DSMT4 is  EMBED Equation.DSMT4 . How can we apply the Chain Rule to functions of the form EMBED Equation.DSMT4 ? To apply the Chain Rule to the function of the form  EMBED Equation.DSMT4 , we use the following formulas: ( Table 5.6 The Chain Rule for Logarithmic Functions If  EMBED Equation.DSMT4 exists, and EMBED Equation.DSMT4 , then  EMBED Equation.DSMT4 If  EMBED Equation.DSMT4 exists, and EMBED Equation.DSMT4  with EMBED Equation.DSMT4 , then  EMBED Equation.DSMT4  Note that the derivative of  EMBED Equation.DSMT4  is a special case of the derivative of  EMBED Equation.DSMT4  because  EMBED Equation.DSMT4 . ( Example 5.14 Determine the derivative for the following functions. (a)  EMBED Equation.DSMT4  (b)  EMBED Equation.DSMT4  (c)  EMBED Equation.DSMT4  ( Solution (a) In this function,  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 , then by the Chain Rule,   EMBED Equation.DSMT4    (b) Here, we need to use The Chain Rule twice. Let  EMBED Equation.DSMT4 , then   EMBED Equation.DSMT4  Now, applying the Chain Rule again, we have  EMBED Equation.DSMT4  We can also find  EMBED Equation.DSMT4 by first rewriting the function using the logarithmic property, and then taking the derivative:   EMBED Equation.DSMT4   EMBED Equation.DSMT4  (c) if  EMBED Equation.DSMT4 , then we use the Quotient Rule to find  EMBED Equation.DSMT4 .   EMBED Equation.DSMT4  Another way of finding EMBED Equation.DSMT4 is by rewriting the function using the logarithmic property, and then taking the derivative:   EMBED Equation.DSMT4  To get the same answer, we need to combine the result as follows:  EMBED Equation.DSMT4  ( ( Example 5.15 Suppose that the demand equation of a product is given by EMBED Equation.DSMT4 , where x is measured in thousands. Find the marginal revenue function and find where the marginal revenue is zero. ( Solution Recall that the revenue function is defined as  EMBED Equation.DSMT4  and the marginal revenue function is EMBED Equation.DSMT4 . Thus,   EMBED Equation.DSMT4  To find where the marginal revenue is zero, let  EMBED Equation.DSMT4 . Thus  EMBED Equation.DSMT4  Recall that  EMBED Equation.DSMT4 is never zero no matter what the value of x is. Thus, the marginal revenue will be zero only if x = 3 or x = -3. For this case, x = -3 doesnt make sense. Therefore, the marginal revenue will be zero if three thousand products are made. ( ( Example 5.16 Find where the tangent line to the graph  EMBED Equation.DSMT4  is horizontal. ( Solution The tangent line will be horizontal when the slope is zero, that is, at points where the derivative is zero. Applying the Chain Rule, we have  EMBED Equation.DSMT4  Since the only time that a fraction can equal zero is when the numerator equals zero,  EMBED Equation.DSMT4 . By solving this equation, we find that the graph has two horizontal tangent lines at x = 0 and x = . ( 5.4 Higher-Order Derivatives The derivative EMBED Equation.DSMT4 of a function f is also a function; hence, where it exists, we can also find the derivative of  EMBED Equation.DSMT4 . We call  EMBED Equation.DSMT4  the first order derivative of f , and the derivative of  EMBED Equation.DSMT4 , denoted as  EMBED Equation.DSMT4 , as the second order derivative of f. Continuing in this manner, we are led to consider the third, fourth, and higher order derivatives of f whenever they exist. The following table represents the notations for the higher order derivatives. ( Table 5.7 Notations for the Higher-Order Derivatives NameNotationThe First-Order Derivative  EMBED Equation.DSMT4 The Second-Order Derivative EMBED Equation.DSMT4 The Third-Order Derivative  EMBED Equation.DSMT4 The  EMBED Equation.DSMT4 -Order Derivative ( EMBED Equation.DSMT4 ) EMBED Equation.DSMT4 Assume EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 exist.  ( Example 5.17 Find all derivatives of all orders of the function  EMBED Equation.DSMT4 . ( Solution We have  EMBED Equation.DSMT4  In fact  EMBED Equation.DSMT4 , for all  EMBED Equation.DSMT4 . ( ( Example 5.18 Find the fourth derivative of the function  EMBED Equation.DSMT4 . ( Solution We have  EMBED Equation.DSMT4   EMBED Equation.DSMT4  ( Application What exactly does the second derivative EMBED Equation.DSMT4 tell us? Just as the first derivative EMBED Equation.DSMT4 measures the instantaneous rate of change of the function f , the second derivative of f measures the rate of change of the first derivative EMBED Equation.DSMT4 of the function f; in other words, the rate of change of the rate of change of f. Acceleration In physics, the second derivative  EMBED Equation.DSMT4  can be interpreted as acceleration. Recall that if  EMBED Equation.DSMT4 is the position function of an object that moves in a straight line, then the instantaneous rate of change of the position of the object is called the velocity of the object at time t:  EMBED Equation.DSMT4  The instantaneous rate of change of velocity with respect to time is called the acceleration a(t) of the object. Thus, the acceleration function is the derivative of the velocity function, and hence, is the second derivative of the position function:  EMBED Equation.DSMT4  ( Example 5.19 The vertical position of a ball thrown from the top of a lighthouse is given by  EMBED Equation.DSMT4  where s is measured in feet and t is measured in seconds. Find the velocity and acceleration of the ball 5 seconds after it is thrown. ( Solution The velocity v and acceleration a of the ball at any time t are given by  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  Therefore, the velocity and the acceleration of the ball 5 seconds after it is thrown is  EMBED Equation.DSMT4   EMBED Equation.DSMT4  This means that after 5 seconds the ball is falling at a speed of 64 ft/sec and the acceleration of the ball is a constant  EMBED Equation.DSMT4 . The negative sign indicates that the acceleration is also directed downward. We should not be surprised by the constant acceleration since we know there is only one force acting on such a following object, gravity, which is constant. ( Concavity The second derivative  EMBED Equation.DSMT4  can be used to study the shape of the graph of a function. For example both  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4  are increasing in the interval (0, "), but there is a fundamental difference in the way they are increasing. The slopes of the lines tangent to the graph of EMBED Equation.DSMT4 are increasing as we move from left to right. Meanwhile, the slopes of lines tangent to the graph  EMBED Equation.DSMT4 are decreasing as we move from left to right. See the following graphs. ( Figure 5.2 Graphs of EMBED Equation.DSMT4  and  EMBED Equation.DSMT4     The word concavity is used to describe the shape of a function. From the above graphs, we have the following definition of concavity. ( Table 5.8 Definition of Concavity We say that the graph of f is concave up on the interval (a, b), if  EMBED Equation.DSMT4 is increasing on (a, b). We say that the graph of f is concave down on the interval (a, b), if EMBED Equation.DSMT4 is decreasing on (a, b).A point where the graph of f changes concavity is called a point of inflection.  Recall that  EMBED Equation.DSMT4  tells us where f is increasing and where it is decreasing. The derivative of  EMBED Equation.DSMT4  , which is  EMBED Equation.DSMT4 , tells us where  EMBED Equation.DSMT4  is increasing and where it is decreasing. We can use the following second derivative test for concavity. ( Table 5.9 Test for Concavity If  EMBED Equation.DSMT4 on the interval (a, b) ( EMBED Equation.DSMT4 is increasing) , then f is concave up on (a, b). If  EMBED Equation.DSMT4 on the interval (a, b) ( EMBED Equation.DSMT4 is decreasing) , then f is concave down on (a, b).  Concavity is very important for sketching the graph of a function, which we discussed it in Chapter 6. Here we will concentrate on its application to economics problems. ( Example 5.20 The total sales (in thousands) of a new video game is given by  EMBED Equation.DSMT4  where t is the time in months. Where is the graph of  EMBED Equation.DSMT4  concave up and where is it concave down? Are there any inflection points of  EMBED Equation.DSMT4 ? Explain what this means. ( Solution The given expression s(t) is called the logistic function. A logistic function is always increasing. Its graph is concave up at first, then concave down, and finally levels off at a horizontal asymptote. (See Figure 5.3). In this case, the point of inflection (also called the point of diminishing), where the concavity changes, is somewhere around 10 weeks. To find out exactly where the curve change the concavity, we need the second derivative of s(t). ( Figure 5.3 Graph of the logistic function   When the curve is concave up, we know EMBED Equation.DSMT4 ( EMBED Equation.DSMT4 is increasing), and when the curve is concave down,  EMBED Equation.DSMT4 ( EMBED Equation.DSMT4 is decreasing). Therefore, the curve changes concavity when  EMBED Equation.DSMT4 or when EMBED Equation.DSMT4  does not exist. Since  EMBED Equation.DSMT4  We can use the chain rule or quotient rule to find the first derivative of s(t):  EMBED Equation.DSMT4  Using the quotient rule and the chain rule, we have  EMBED Equation.DSMT4  The denominator in  EMBED Equation.DSMT4  is never zero, so there is no point at which EMBED Equation.DSMT4 is undefined. To find where the  EMBED Equation.DSMT4 , we only need to set the numerator equals to zero:  EMBED Equation.DSMT4  The factor  EMBED Equation.DSMT4  can never be equal to zero, so we have  EMBED Equation.DSMT4 . Now, solve for t:   EMBED Equation.DSMT4  Thus, the curve of the sales function changes the concavity around 10.4 months. This means when the new game just appears on the market, the sales increase rapidly, more units are sold each month than the previous month until the middle of the 11th month. Then the sales slow down and fewer units are sold each month than the previous month. The inflection point is the time when the rate of sales stops increasing. Eventually, most people who want the new game have already bought it, the sales will stop and the 59 million units should be the maximum potential sales of the new game. ( Sample Quiz Find derivatives for problems 1- 5: 1.  EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4  3.  EMBED Equation.DSMT4  4.  EMBED Equation.DSMT4  5.  EMBED Equation.DSMT4  6. Find the equation of the tangent line to the graph of  EMBED Equation.DSMT4 at the point at which x = 0. 7. The cost function for a product is given by  EMBED Equation.DSMT4 , where x is the number of items produced and C(x) is the cost in dollars to produce x items. How much would it cost to produce the 101st item? 8. A drug concentration curve is given by  EMBED Equation.DSMT4 , with C in mg/ml and t in minutes. What is the rate of the concentration 4 minutes after it was administered? 9. Suppose the demand equation for a product is given by  EMBED Equation.DSMT4 , where x is the number of items sold and p is the price in dollars. Find the marginal revenue function. 10. The deer population in a wild park is approximated by  EMBED Equation.DSMT4  with t measured in years since 1980. When does the deer population stop growing? Answers to Sample Quiz 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. $100.6 8. 1.797 mg/ml/min 9.  EMBED Equation.DSMT4  10. After t = 7.324 years or in 1987 Recall that:  EMBED Equation.DSMT4  Apply the Chain Rule for each term. Recall that:  EMBED Equation.DSMT4  Apply the Chain Rule for  EMBED Equation.DSMT4  The General Power Rule Factor out the common factor Simplify Distributive Property Simplify. Factor out the common factor  EMBED Equation.DSMT4  from the numerator. Simplify. Apply the Quotient Rule. Apply the Chain Rule to each term. Factor out the common factor:  EMBED Equation.DSMT4  Simplify. Apply the Product Rule. Apply the General Power Rule for derivatives. Simplify. Factor the common factor:  EMBED Equation.DSMT4  Apply the Distributive property. Simplify. Rewrite General Power Rule Rewrite the outer function as  EMBED Equation.DSMT4  General Power Rule Simply Rewrite  EMBED Equation.DSMT4  as  EMBED Equation.DSMT4  n  EMBED Equation.DSMT4   EMBED Equation.DSMT4  Replacing u by EMBED Equation.DSMT4 . These steps can be performed mentally. The derivative of the outer function EMBED Equation.DSMT4 , where  EMBED Equation.DSMT4  Derivative of the inner function at x Derivative of the outer function at u = g(x) Note that when taking the derivative of the outer function, do not change the inner function g(x). The derivative of the inner function g(x) The derivative of the outer function  EMBED Equation.DSMT4 , where  EMBED Equation.DSMT4  The derivative of inner function  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  Derivative of inner function g at x Derivative of outer function f at  EMBED Equation.DSMT4  Inner function Outer function Rewrite  EMBED Equation.DSMT4 as  EMBED Equation.DSMT4 . Power rule and simplify.  EMBED Equation.DSMT4  (1, 1) Product Rule Distributive property Simplify. Sum and Difference rules Rule 3 Power Rule and Rule 1 Simplify. Apply the Sum, Difference, and Power rules. Note that  EMBED Equation.DSMT4 is a constant.  EMBED Equation.DSMT4   EMBED Equation.DSMT4 is a constant, it does not need to be rewritten. Apply the Sum, Difference and Power rules. Simplify. Note:  EMBED Equation.DSMT4   EMBED Equation.DSMT4  Apply the quotient rule for g(x). Simplify Point of inflection point. The rate of sales stop increasing. Concave up. The rate of sales is increasing. More units are being sold each month. Maximum potential sales Concave down. The rate of sales is decreasing. Less units are being sold each moth. The exponential function with base e. Taking the nature logarithm Slope is decreasing means  EMBED Equation.DSMT4  is decreasing. We say f is concave down. Slope is increasing means  EMBED Equation.DSMT4 is increasing. 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