ࡱ> EGkR dbjbj 7nxxتDDD%D%D%D%DF%DN.TTTT__ _֓ؓؓؓؓؓؓ$GxED``T& _``TTeA`n!TDT֓`֓37 T{,%DbZ“W0a~PaaD_>_,`$B`___B ___````a_________ : AP Calculus Assignments: Derivative Concepts DayTopicAssignment1Average Rate of ChangeHW Derivative Concepts - 1 2Slope of a CurveHW Derivative Concepts - 2 3The derivative at a point; nDerivHW Derivative Concepts - 34The derivative functionHW Derivative Concepts - 4 5Differentiability and continuityHW Derivative Concepts - 56Increasing/decreasing functions, stationary pointsHW Derivative Concepts - 67Concavity, inflection pointsHW Derivative Concepts - 78Practice with graphs **QUIZ**HW Derivative Concepts - 8 9Estimating derivatives from tables and graphsHW Derivative Concepts - 9 10Derivative interpretationHW Derivative Concepts - 10 11Review and practice **QUIZ**HW Derivative Concepts 1112ReviewHW Derivative Concepts - Review13**TEST** AP Calculus HW: Derivative Concepts - 1 1. a. Write the formula for average rate of change of a function f over the interval [a, b]. b. Give the geometric (graphical) interpretation of rate of change. c. Memorize both of the above. 2. Find the average rate of change of  EMBED Equation.DSMT4  on the interval [1, 16]. 3. The average rate of change of a function on the interval [2, 6] is 1.75. If (6) = 3, find (2). 4. The average rate of change of a function on the interval [a, b] is k. Find an expression for f(b) in terms of a, b, k and f(a). 5. The average rate of change of a function f on the interval [0, 6] is 3 and the average rate of change on the interval [6, 10] is 2. What is the average rate of change on the interval [0, 10]? 6. If the average rate of change of a function f on the interval [2, 5] is 3 and f(2) = 1, which of the following must be true? a. f(5) > f(2) b. f is increasing on [2, 5] c. f does not have a root in [2, 5] d. f(3) = 4 e. f(5) = 10 f. The equation of the secant line to f on the interval [2, 5] is y = 3x 5. 7. What must be true if the average rate of f on [a, b] is 0? 8. Approximately a billion years ago, an alien spacecraft crash-landed on Mars. The surviving aliens were never able to leave but managed to salvage enough materials from their ship to form a small colony. The population of the colony over time is given in the table below. Years since crash 50100150200250300350400450Population137162187212236259272254150 a. Find the average rate of change of the colonys population between the years 50 and 300. b. Write a linear equation to describe the alien population for the years 50 to 300. c. Using your equation from part (b), estimate the number of aliens that survived the crash of the ship.  9. The graph of Isabuggas walk from todays notes is shown at right. a. What do we mean by Izzys instantaneous velocity? b. At time t = 10, was Izzys velocity greater than or less than 25 cm/min? How do you know? c. At time t = 21.333, was Izzys velocity greater than or less than 25 cm/min? d. At time t = 29, was Izzys velocity greater than or less than 25 cm/min? e. Was Izzy traveling faster at time t = 12.5 or t = 17.5? (This assignment continued on the next page.) 10. When a model rocket is launched, it burns propellant for a brief time, accelerating it upward. Several seconds after burnout, a parachute opens to slow the rockets fall. The graph of the rockets velocity as a function of time is shown at right. a. How long did the rocket engine burn? b. What was the rockets average acceleration (rate of change of velocity) during the burn stage? c. Describe what happened between burnout and the opening of the parachute. d. What was the rockets average acceleration during the time between burnout and the opening of the parachute? Is this what you would expect? Why? e. When did the rocket reach its maximum height above the ground? 11. On a quiz, Kenny needed to solve the equation x3 + 4x = 7. He tried to do it algebraically. What happened? AP Calculus HW: Derivative Concepts - 2 1. Use your calculator to approximate the slope of  EMBED Equation.DSMT4  at the point x = 2 and write an equation of the tangent line to the graph of f at that point. 2. A certain function f has f(4) = 3 and the slope of the tangent line to f at x = 4 is 1.25. a. Write the equation of the tangent line to the graph of f at x = 4. b. Use the tangent line equation to estimate the value of f(3.7).  3. The graph at right shows the position of an object moving along the x-axis as a function of time. Use the graph to answer the following. a. Write the points A F in order from least to greatest velocity. b. Write the points A F in order from least to greatest speed. c Was the velocity increasing or decreasing at A? At C? At F? d. At what point does the velocity appear to change from increasing to decreasing? From decreasing to increasing?  4. The graph at right shows the vertical position of an elevator as a function of time. a. Describe what happens. b. What was the elevator's instantaneous velocity at t = 10 seconds? c. Sketch a velocity versus time graph for the motion of the elevator for 0 ( t ( 20.  5. The graph at right shows the position versus time curve for a particle moving along a straight line. Use the graph to estimate the following. a. What is the average velocity of the particle 1) on the interval 0 ( t ( 3? 2) on the interval 2 ( t ( 4? 2) on the interval 4 ( t ( 8? b. At what times does the particle have instantaneous velocity of zero? c. When is the instantaneous velocity of the particle 1) a maximum? 2) a minimum?  6. For each of the following graphs of, sketch a rough graph of what the slope of the function should look like. a. b. c. 7. Kenny was given the graph from problem number 5 above and asked whether the particles velocity was greater at time t = 3 or at time t = 5. a. Kenny first said at t = 3 because the y-value is higher there. What happened? b. Kenny then said at t = 3 because the graph is steeper there. What happened? AP Calculus HW: Derivative Concepts - 3 1. Find each derivative without using your calculator; then use nDeriv to check your answer. a. Find f '(2) for (x) = x3. b. Find f '(3) for (x) =  EMBED Equation.DSMT4 . c. Find f '(4) for (x) =  EMBED Equation.DSMT4 . 2. Each limit represents the derivative of some function f at some number a. Find the function f and the number a for each of the following: a.  EMBED Equation.DSMT4  b.  EMBED Equation.DSMT4  c.  EMBED Equation.DSMT4  d.  EMBED Equation.DSMT4  3. A bungee jumper jumps off a 110 foot tower and bounces at the end of her tether. Her height is approximated by h = 50 + 60 e0.2tcos (1.257 t) where h is in feet and t is time in seconds since she jumped. a. Find the jumper's velocity one second after she jumps. (Do it the easy way!) b. Find her velocity three seconds after she jumps. c. What do the signs in parts a and b indicate? 4. a. For the function (x) = x2 4x, find the derivative at an arbitrary point x = a; in other words, find  EMBED Equation.DSMT4 . b. Replace a by x in your answer from part a above and graph both (x) and  EMBED Equation.DSMT4  on the same set of axes. c. Is  EMBED Equation.DSMT4  a function of x? How do you know? d. If  EMBED Equation.DSMT4  represents the slope of , does the graph of  EMBED Equation.DSMT4  "agree" with the graph of ? 5. Kenny couldnt believe his teacher was serious about memorizing the formulas for the derivative so he didnt bother. What happened? AP Calculus HW: Derivative Concepts - 4  1. Match the graph of each function (a) (d) with the graph of its derivative (1) (4) below. (a) (b) (c) (d) (1) (2) (3) (4) 2. For each of the functions shown, sketch a graph of the derivative.  a. b. c. d. 3. Find the derivative of the given function using the definition of the derivative. a.  EMBED Equation.DSMT4  b.  EMBED Equation.DSMT4  4. Recall the three general ways a function can be discontinuous. At a point of discontinuity, can a function have a derivative? Explain or illustrate. 5. Graph the function (x) = |x 2| + 1. a. Is the function continuous at x = 2? b. Does the function have a derivative at x = 2? Explain. 6. Graph the function (x) =  EQ \r(3,x 2) . Does it have a derivative at x = 2? Explain. Hint: it may help to zoom in several times on the point (2, 0).  7. Kenny was asked about the derivative of the function shown at right. Kenny said the derivative was negative and increasing. What happened? AP Calculus HW: Derivative Concepts - 5 1. Briefly (no more than two words each) identify two ways a continuous function could fail to be differentiable at a point. Illustrate each with a diagram.  2. A function f is graphed at right. Note that f has a horizontal tangent at x = 7 and a vertical tangent at x = 9. Name the x-values where a. f is discontinuous. b. f is not differentiable.  3. Consider the function  EMBED Equation.DSMT4 . Is f continuous at x = 0? Justify your answer.  4. Consider the function  EMBED Equation.DSMT4 . Is f continuous at x = 0? Justify your answer. 5. Read Increasing and Decreasing Functions and Derivatives which is at the end of the HW packet. (If you chose not to read this, try not to ask questions next class that make it appear that you are lazy and/or illiterate.) Then, as a preview of coming attractions, answer the following. The function  EMBED Equation.DSMT4  is continuous for all x. f '(x) < 0 on (((, (2) and (2, () and f '(x) > 0 on ((2, 2). a. On what interval(s) is f increasing? Justify your answer. b. On what interval(s) is f decreasing? Justify your answer. 6. Kenny was given a function g that was continuous at the point x = 7. Kenny concluded that g'(7) must exist. What happened? AP Calculus HW: Derivative Concepts 6  1. The derivative f ' of a function f , with domain [0, 10] is graphed at right. Use the graph to answer the following: a. On what intervals is f increasing? decreasing? b. Where is f increasing the fastest? decreasing the fastest? c. At what points is f stationary? d. Classify the stationary points as relative maxima, relative minima, or neither. (This assignment continued on the next page.)  2. The derivative  EMBED Equation.DSMT4  of a function g with domain [0, 8] is graphed at right. From the graph, determine a. On what interval(s) is g increasing? Decreasing? b. Which is greater g(0) or g(1)? Explain c. Which is greater, g(3) or g(6)? Explain. d. Where does g have a relative minimum? Explain. e. Where does g have a relative maximum? Explain. f. Find an upper and lower bound for the difference g(5) g(4). 3. The derivative of a continuous function f is given by  EMBED Equation.DSMT4 . a. Find the critical points of f. b. On what interval(s) is f increasing? decreasing? Justify your answers. 4. The derivative of a continuous function f is given by f '(x) = x3 x2 6x. a. Find the critical points of f. b. On what interval(s) is f increasing? Decreasing? Justify your answers. 5. For each derivative graphed below draw a possible graph of the original function .  a. b. c. 6. Sketch a graph of the derivative of each of the following functions.  a. b. c. 7. Kenny kept mixing up the words increasing and positive. Afterward, hed tell the teacher You knew what I meant. What happened? AP Calculus HW: Derivative Concepts 7 1. For each function graphed below, give the signs of both the first and second derivatives.  a. b. c. d. e. f.  2. The graph of the derivative  EMBED Equation.DSMT4  of a function is given at right. From the graph, determine on what interval(s) the graph of is concave up and on what interval(s) the graph is concave down.  3. The graph at left represents the second derivative of a function g. From the graph, determine a. On what interval(s) is the graph of g concave up? concave down?  b. Where does the graph of g have points of inflection? 4. In the diagrams a d below represent the graphs of four different functions while (1) (4) represent the second derivatives of those functions. Match each graph with its correct second derivative.  a. b. c. d. (1) (2) (3) (4) (This assignment continued on the next page.) 5. A continuous function f has first derivative  EMBED Equation.DSMT4  and second derivative  EMBED Equation.DSMT4 . a. Determine the intervals on which f is increasing and decreasing and find and classify the relative extrema of f. b. Determine the intervals on which f is concave up and concave down and find the inflection points of f. 6. a. Sketch a graph of a function that is concave up and draw the tangent line to the graph at an arbitrary point. b. If the tangent line is used to estimate function values near the point of tangency, will the estimates be over or under estimates? c. Does the answer to part (b) above depend on whether the function is increasing or decreasing? d. If the graph of a function is concave down and the tangent line is used to estimate function values near the point of tangency, will the estimates be over or under estimates? 7. You may already know this from Physics. If you do, great. If you dont, learn it. Velocity is a vector; it has a size and a direction. For motion along a line, this means that velocity is a signed quantity. v = 2 m/s means an object is moving 2 meters per second to the right (or up); v = 2 m/s means the object is moving 2 meters per second to the left (or down). Velocity is the rate of change (derivative) of position. Speed is a scalar; it has (non-negative) size but no direction. A speed of 2 m/s means an object is moving 2 meters per second but does not tell which way. Speed = |v|. For each of the position time graphs shown below, tell 1) whether the velocity is increasing or decreasing at the point indicated on the graph and 2) whether the speed is increasing or decreasing at the same point.  a. b. c. d.  8. Kenny was asked what he could say about the slope and the derivative of the function shown. He said The slope is increasing because the derivative is positive. What happened? AP Calculus HW: Derivative Concepts - 8  1. Match each of the 12 graphs in a - l with the correct graph A - L of its derivative. a. b. c. d.  e. f. g. h.  i. j. k. l.  A. B. C. D.  E. F. G. H.  I. J. K. L. (This assignment continued on the next page.) AP Calculus HW: Derivative Concepts - 8 (continued) 2. The graph of a function is shown. Sketch the graphs of both ' and ".   a. b.   3. The graphs below show the position, velocity, and acceleration of a car as a function of time in minutes. Identify which is which. Note: velocity is the rate of change of position and acceleration is the rate of change of velocity.   4. The graph at right shows the derivative of a function f. Based on the graph, a. On what intervals is increasing? b. On what intervals is concave up? c. Where does have a relative maximum? relative minimum? d. Which is greater, (2) or (3)? Explain. e. If f(2) = 8, between what two integers is the value of f(3)? Explain.  f. If (3) = 5, between what two integers is the value of (2)? Explain. 5. Kenny was given the graph of the derivative of a function, f '. Unfortunately, he didnt pay attention and thought it was a graph of the function f. What happened? AP Calculus HW: Derivative Concepts - 9 1. A high performance sports car accelerates from 0 to 60 mph in 5 seconds. Its velocity (in ft/s) is given at one-second intervals in the table below. Time (s)012345Velocity (ft/s)03052688088 Estimate the acceleration of the car at a. t = 2, b. t = 0, and c. t = 5 seconds. 2. The table below gives the life expectancy E of men born in the US during certain years t. Year (t)1900191019201930194019501960197019801990Life Expectancy (E) 48.351.155.257.462.565.666.667.170.071.8 Use the table to estimate the values of. a. E' (1950) b. E' (1900) c. E' (1990) d. E' (1935) 3. A skydiver jumps from a plane. During freefall (before she opens her 'chute) the distance she falls in feet as a function of time in seconds is given by s(t) = 176t + 986[(.835)t 1] 0 ( t ( 20 a. What is her average velocity for the first 15 seconds of freefall? b. What is her instantaneous velocity after 15 seconds.  4. The graph at right shows the position in meters as a function of time in seconds of an object moving along the x-axis. Use the graph to estimate the following. a. The object's velocity at t = 4. b. The object's velocity at t = 14. c. The object's maximum velocity. d. The object's acceleration at t = 5. 5. A frustrated driver is caught behind a slow moving farm vehicle. At time t = 0, he sees his chance, pulls into the passing lane, and stomps on the gas. His average velocity in ft/s in the interval from t = 1 sec to t = 1 + h seconds is given by vavg =  EMBED Equation.3 . Estimate the irate driver's instantaneous speed at t = 1s.  6. Kenny was asked to estimate the value of f '(15) from the table shown. Kenny got  EMBED Equation.DSMT4 . What happened? AP Calculus HW: Derivative Concepts - 10 1. Let T(t) describe the temperature in degrees Fahrenheit of a potato t minutes after it is placed in a hot oven. a. What are the sign and the units of T'(t)? b. In practical terms, what does the equation T '(20) = 2 mean? 2. The cost in dollars of drilling a railroad tunnel through solid rock is estimated to be c = f(x) where x is the length of the tunnel in feet. a. What are the units of  EMBED Equation.DSMT4 ? b. What does  EMBED Equation.DSMT4  represent in practical terms? c. What is the sign of  EMBED Equation.DSMT4  ? d. If  EMBED Equation.DSMT4  = $3000 when x = 1500', estimate the cost of drilling the tunnel two feet longer. 3. The ACME Widget Co. estimates that it can sell N = (p) widgets/year at a price of p dollars per widget. a. What is the practical meaning of (20) = 50000? What are the units of the 20? The 50000? b. What is the sign of  EMBED Equation.DSMT4 ? Explain. c. What is the practical meaning of '(20) = 2000? What are the units of the 2000? d. Using (20) = 50000 and '(20) = 2000, estimate the effect of a price change to 1) p = $21? 2) p = $19.50? 4. At time t = 0 minutes, Lara begins running water for a bath. Let H(t) represent the depth of the water in the bathtub after t minutes. Lara runs the water for 5 minutes, spends two minutes looking for her favorite bath toy before climbing into the tub and then plays for 10 minutes before pulling the plug to let the water drain, which takes three minutes. a. Draw a graph of H(t). b. Draw a graph of H '(t). c. What are the units of H '(t)? 5. Kenny had a homework problem about water flowing into a conical paper cup. Kenny was told that at time t = 5 seconds, the depth of the water in the cup, h, in inches, was changing at a rate of  EMBED Equation.DSMT4 in/sec and asked what he could say about the depth of water in the cup at that instant. Kenny concluded that  EMBED Equation.DSMT4 ( h = 0.4t so after 5 seconds, the cup had 0.4(5) = 2 inches of water in it. What happened? AP Calculus: Derivative Concepts - 11  1. The graph of g' is shown at right. Which of the following statements is/are true of g at x = a? I. g is continuous II. g is differentiable III. g is increasing (A) I only (B) III only (C) I and III only (D) II and III only (E) I, II and III 2. Given that f is a function, how many of the following statements are true? (i) If f is continuous at x = c, then  EMBED Equation.DSMT4  exists. (ii) If  EMBED Equation.DSMT4  exists, then f is continuous at x = c. (iii)  EMBED Equation.DSMT4  (iv) If f is continuous on (a, b), then f is continuous on [a, b]. (A) 0 (B) 1 (C) 2 (D) 3 (E) 4  Use the graph of f at right to answer questions 3 6. 3. Over which of the following intervals does  EMBED Equation.DSMT4  equal 0? I. (6, 3) II. (3, 1) III. (2, 5) (A) I only (B) II only (C) I and II only (D) I and III only (E) II and III only 4. How many points of discontinuity does  EMBED Equation.DSMT4  have on the interval 6 < x < 7? (A) none (B) 2 (C) 3 (D) 4 (E) 5 5. For 6 < x < 3,  EMBED Equation.DSMT4  equals (A) 3/2 (B) 1 (C) 1 (D) 3/2 (E) 2 6. Which of the following statements about the graph of  EMBED Equation.DSMT4  is false? (A) It consists of six horizontal segments (B) It has exactly four jump discontinuities (C)  EMBED Equation.DSMT4  is discontinuous at each x in the set {3, 1, 1, 2, 5} (D)  EMBED Equation.DSMT4  ranges from 3 to 2 (E) On the interval 1 < x < 1,  EMBED Equation.DSMT4  = 3. (This assignment continued on the next page.) Use the graph at right to answer questions 7 8. The graph consists of two line segments and a semicircle. 7.  EMBED Equation.DSMT4  for x = (A) 1 only (B) 2 only (C) 4 only (D) 1 and 4 (E) 2 and 6 8.  EMBED Equation.DSMT4 does not exist for x = (A) 1 only (B) 2 only (C) 1 and 2 only (D) 2 and 6 only (E) 1, 2 and 6 9. The table gives the vales of a function f that is differentiable on the interval [0, 1] x0.10.20.30.40.50.6f(x)0.1710.2880.3570.3840.3750.366 The best approximation of  EMBED Equation.DSMT4  according to this table is (A) 0.12 (B) 1.08 (C) 1.17 (D) 1.77 (E) 2.88 Use the graph of y = f(x) shown at right to answer questions 15 and 16.  10.  EMBED Equation.DSMT4  is most closely approximated by (A) 0.3 (B) 0.8 (C) 1.5 (D) 1.8 (E) 2 11. The rate of change of f is least at x ( (A) 3.0 (B) 1.3 (C) 0.5 (D) 2.7 (E) 4.0 (This assignment continued on the next page.) 12. A function f has the derivative shown at right. Which of the following statements must be false? (A) f is continuous at x = a (B) f(a) = 0 (C) f has a vertical asymptote at x = a (D) f has a jump discontinuity at x = a (E) f has a removable discontinuity at x = a 13. The function f whose graph is shown at right has f ' = 0 at x = (A) 2 only (B) 2 and 5 (C) 4 and 7 (D) 2, 4 and 7 (E) 2, 4, 5, and 7 14. A differentiable function f has the values shown. Which of the following is the best estimate of f '(1.5)? x1.01.21.41.6f(x)8101422 (A) 8 (B) 12 (C) 18 (D) 40 (E) 80 15. Kenny believed that because multiple choice problems did not require work to be shown, it meant that no work had to be done (all problems could be done mentally). What happened? (A) Kenny died horribly. (B) Kenny died horribly. (C) Kenny died horribly. (D) Kenny died horribly. (E) All of the above. AP Calculus Review: Derivative Concepts 1. a. Write both forms of the definition of a derivative at a point a. b. Give two interpretations of the derivative of a function at a point a. c. Give two reasons why a continuous function might fail to have a derivative at a point a. 2. Each of the limits below represents '(a) for some function f and some number a. Find (x) and a for each. a.  EMBED Equation.DSMT4  b.  EMBED Equation.DSMT4  c.  EMBED Equation.DSMT4  d.  EMBED Equation.DSMT4   3. A function f is graphed at right. F has a vertical tangent at x = 2 and horizontal tangents at x = 3, x = 1 and x = 3. a. Where does f have discontinuities? b. At what points does f fail to be differentiable? c. Make a sketch of the derivative of f. 4. Sketch the graph of differentiable functions , g and h that satisfy the given conditions: a. (0) = 1,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4  for x < 0, and  EMBED Equation.DSMT4  for x > 0. b. g(x) > 0 for all x and  EMBED Equation.3  for all x. c. h(x) > 0,  EMBED Equation.3 , and  EMBED Equation.3  for all x > 0. 5. $1000 is compounded continuously at a rate of r%. After 10 years the balance is B where B = (r). a. What does the statement (5) = 1649 mean in the context of the problem? b. What does the statement  EMBED Equation.DSMT4  mean in the context of the problem? c. What are the units of  EMBED Equation.DSMT4 ? 6. a. How is the derivative of f related to intervals where f is increasing? Is this a biconditional? b. How can we find a relative maximum using the derivative? c. What does concave up mean? How are derivatives related to intervals where f is concave up? d. How can we find inflection points of f?   7. The graph of the derivative of a continuous function f is shown at right. a. Find and classify the relative extrema of f. b. Where does f have inflection points? c. On what interval(s) is f increasing? d. On what interval(s) is f concave up? e. What happens to f at x = 1? f. Which is greater: f(2) or f(3)? Why? g. Which is greater: f(5) or f(4)? Why? h. If f(4) = 6, estimate f(5) and f(3). i. If f(1) = 1, estimate f(3). (This assignment continued on the next page.) 8. A rocket powered car is being tested in the California desert. Starting at rest, the car accelerates to the velocities shown in the table where t is in seconds since the car started and v is in meters per second. t (seconds)123510203040v (meters per second)2168135186214226228230 a. Estimate the cars acceleration at time t = 2 seconds. Give units. b. Estimate the cars acceleration at time t = 40 seconds. c. Write a linear equation to estimate the cars velocity for times t > 40 seconds if it continues to accelerate at the same rate. d. Suppose the car fails catastrophically (it basically flies apart) at velocities greater than 250 m/s. What is the last time when the driver can safely eject? e. Estimate how many meters the car travels in the interval 30 ( t ( 40 seconds. 9. When a skydiver jumps out of a plane, two main forces are affecting her speed. Gravity tries to speed her up, while air resistance tries to slow her down. The effect of gravity is (roughly) constant while the effect of air resistance increases at greater speeds. Suppose v(t) represents the skydivers speed in feet per second (measured downward so v > 0) as a function of time t in seconds since she left the plane. a. What is the sign of v' ? Justify your answer. b. If v(5) = 150 and |v'(5)| = 12, estimate v(7). b. What is the sign of v" ? Justify your answer. c. Is your estimate of v(7) from part (b) an overestimate or an underestimate? Justify your answer. 10. A driver is accelerating on the highway. His speed in feet per second at three different times (in seconds) is given by v(4) = 82.400, v(4.01) = 82.423 and v(7) = 88.250. a. What was his average acceleration on the interval 4 ( t ( 7 seconds? Include units. b. Estimate his instantaneous acceleration at t = 4. c. If v" does not change sign during the interval 4 ( t ( 7, what is its sign? Justify your answer. 11. Kenny didnt really understand a lot of the stuff in this unit but figured he could afford one poor test grade and the course would move on to other topics. What happened? Answers to selected problems HW - 1 1a.  EMBED Equation.DSMT4  b. The slope of the secant line from  EMBED Equation.DSMT4  to  EMBED Equation.DSMT4  2. 2/3 3. 11 4.  EMBED Equation.DSMT4  5. 2.6 6. a, e and f are true 7. f(a) = f(b) 8a. 0.488 aliens/yr b. P = 0.488t + 112.6 c. 112 or 113 9a. Izzys instantaneous velocity is her velocity at a particular instant in time. Izzys instantaneous velocity (usually just called velocity for brevity) is the slope of her position-time graph at a particular time. Well talk more next class about just we mean by the slope of a curve. b. Greater. At t = 10, Izzys position graph is increasing faster than Bufords; it has a greater slope. c. Less. d. Greater. e. 12.5 10a. 2 sec b. 95 ft/s2 c. See solutions d. 31.8 ft/s2 e. 8 sec HW 2 1. m = 0.5; y ( 1 = 0.5(x ( 2) 2a. y 3 = 1.25(x 4) b. 3.375  3a. E, D, F, A, C, B b. F, A, C, D, B, E (possibly open to debate) c. Increasing, decreasing, increasing d. B and E, respectively 4a. Elevator starts at rest for about 2.5 seconds, then rises at a constant speed for 15 seconds, then returns to rest. b. 4 m/s c. See graph at right. 5a1. 0 cm/s a2. -5 cm/s a3. 1.25 cm/s b. At t = 0, 2, 4, and 8 s.  c1. t ( 1 s c2. t ( 3 s 6a. b. c. . 7. Kenny died horribly. Twice in one day. HW 3 1a. 12 b. 1/4 c. 1/2 2a.  EMBED Equation.DSMT4 , a = 1 b.  EMBED Equation.DSMT4 , a = 2 c.  EMBED Equation.DSMT4 , a = 1 d.  EMBED Equation.DSMT4 , a = 0 3a. 61.766 ft/s b. 29.689 ft/s c. Going down at t = 1, up at t = 3 4a.  EMBED Equation.DSMT4  HW 4 1a. (2) b. (4) c. (1) d. (3)  2 a. b. c. d. 3a.  EMBED Equation.DSMT4  b.  EMBED Equation.DSMT4  4. No. 5a. Yes b. No, there is not a unique tangent line at x = 2 6. No, the tangent line is vertical so its slope is undefined HW 5 1. A vertical tangent or a corner. 2a. x = 3, x = 12, x = 14 b. x = 1, x = 3, x = 5, x = 9, x = 12, x = 14 3. No because  EMBED Equation.DSMT4  does not exist. 4. Yes.  EMBED Equation.DSMT4 . 5a. f is increasing on (((, (2) and (2, () because f ' > 0 on those intervals is acceptable for the AP test. f is increasing on (((, (2] and [2, () because f ' > 0 on (((, (2) and (2, () is a better answer (if a little longer). b. f is decreasing on ((2, 2) because f ' < 0 on that interval is acceptable. f is decreasing on [(2, 2] because f ' < 0 on ((2, 2) is better. HW 6 1a. Increasing on [0, 5.5] and [9, 10]; decreasing on [5.5, 9] b. Increasing fastest at x = 4; decreasing fastest at x = 7 c. x = 1, 5.5, and 9 d. x = 1: neither; x = 5.5: relative max; x = 9: relative min 2a. Increasing on [3, 6], decreasing on [0, 3] and [6, 8] b. g(0) c. g(6) d. x = 3 e. x = 6 f.  EMBED Equation.DSMT4  3a. x = 0, x = 5 b. f increasing on [0, 5] b/c f ' > 0 on (0, 5); f decreasing on ((, 0] and [5, () b/c f ' < 0 on ((, 0) and (5, () 4a. x = 2, x = 0 and x = 3 b. f decreasing on ((, 2] and [0, 3] b/c f ' < 0 on ((, 2) and (3, (); f increasing on [2, 0] and [3, () b/c f ' > 0on (2, 0) and (3, () 5a. b. c.  6.  a. b. c. 7. For once, Kenny didnt die. But he kept losing lots of points on his AP Calc quizzes and tests. HW 7  1a. +, + b. 0, 0 c. , 0 d. , + e. +, f. , 2. Concave up on [2, 2] and down on [5, 2] and [2, 5] 3a. Concave up on [3, 6] and down on [0, 3] and [6, 8] b. IPs at x = 3 and x = 6 4a. (4) b. (3) c. (1) d. (2) 5a. f is decreasing on ( ", 6] because f ' < 0 on ( ", 6). f is increasing on [6, ") because f ' > 0 on (6, 8) and (8, "). f has a relative minimum at x = 6 b/c f ' changes from negative to positive there. b. f is concave up on ( ", 8) and (12, ") because f " > 0 on those intervals. f is concave down on (8, 12) because f " < 0 on that interval. f has inflection points at x = 8 and x = 12 because f " changes sign at those points. 6a. See graph at right b. Under. c. No d. Over 7a. Both decreasing b. both increasing c. v incr, speed decr d. v decr, speed incr HW 8  1a. E b. K c. D d. G e. F f. J g. C h. L i. A j. I k. B l. H 2a. b . 3. C is position, A is velocity and B is acceleration 4a. [1, 6] b. [3, 2], [4, 5] c. Relative max at x = 6; relative min at x = 1 d. f(3) since f ' > 0 on (2, 3) e. 10 < f(3) <11 f. 2 < f(2) < 3 HW 9 1a. 19 ft/s2 b. 30 ft/s2 c. 8 ft/s2 2a. 0.205 yr/yr b. 0.280 yr/yr c. 0.180 yr/yr d. 0.510 yr/yr 3a. 114.663 ft/s b. 164.109 ft/s 4a. 10.5 m/s b. 11 m/s c. 24 m/s d. 6.75 m/s2 5. 55.8 ft/s HW 10 1a. + (F/min b. At t = 20 minutes, the potato is heating up at a rate of 2(F/min. 2a. $/ft b. The (instantaneous) cost of digging one more foot. c. + d. $6000 3a. At $20 per widget, ACME expects to sell 50,000 widgets. b. ; As the cost goes up, demand should go down. c. When the price is $20/widget, a dollar increase in price would mean a 2000 widget/yr decrease in sales.  d1. Sales go down to 48,000/yr d2. Sales go up to 51,000/yr 4a. b. c. Inches/minute HW 11 1. E 2. B 3. E 4. E 5. D 6. B 7. C 8. E 9. C 10. B 11. C 12. C 13. A 14. D 15. E Review 1a.  EMBED Equation.DSMT4  b. f '(a) is 1) the instantaneous rate of change of f at x = a and 2) the slope of the tangent line to f at x = a. c. f ' will fail to exist where f has 1) a corner or 2) a vertical tangent  2a. f(x) = x3; a = 4 b.  EMBED Equation.DSMT4 ; a = 3 c. f(x) = cos x; a = 0 d. f(x) = tan x; a = (/4 3a. x = 4, x = 5 b. x = 4, x = 1, x = 2, x = 5  c. See graph at right.  4. a. b.  c.  6a. If f ' > 0 everywhere on an interval (a, b), then f is increasing on [a, b]. The converse is not true. If f is increasing on [a, b], it is possible that f ' = 0 or undefined at a finite number of points on (a, b). b. A continuous function f has a relative maximum where f ' changes from positive to negative. c. A function is concave up where its slope is increasing. If f ' is increasing OR if f " > 0 everywhere on an interval (a, b), then f is concave up on (a, b). d. A continuous function f has an inflection point where f " changes sign. 7a. Relative minimum at  EMBED Equation.DSMT4 ; relative maximum at  EMBED Equation.DSMT4 . b. x = 1, x = 0, x = 3, x = 7 c. (2, 5) and (8.5ish, () (That last one is a wild guesstimate.) d. ((, 1), [0, 3] ( [7, () e. vertical tangent f. f(3) because f ' > 0 on (2, 3) g. f(5) because f ' < 0 on (5, 4) h. f(5) ( 7, f(3) ( 3.5 i. f(3) ( 3 8a. 57 m/s2 b. 0.2 m/s2 c. v = 230 + 0.2(t 40) d. t < 140 sec e. 2290 m 9a. v' > 0 since speed should be increasing. b. 174 ft/s c. Overestimate. As v increases, air resistance increases, so v' will decrease (v will increase more slowly). 10a. 1.95 ft/s2 b. 2.3 ft/s2 c. Negative. Since the average acceleration over 4 ( t ( 7 is less than the instantaneous acceleration at t = 4, the acceleration must be decreasing: v" = a' < 0. 11. Kennys brain liquefied and ran out his ears when he realized that the next two units were all about derivatives and much of the rest of the course was material closely related to derivatives. Increasing and Decreasing Functions and Derivatives We often want to know on what intervals a function is increasing and on what intervals it is decreasing. For the most part, this is not confusing. Everyone agrees that the function EMBED Equation.DSMT4  is increasing on the interval (0, () and decreasing on (((, 0). (Remember, when we talk about increasing and decreasing, we are always going from left to right.) This turns out to be all you really need to know for the AP exam. However, it isnt everything to know. Arguments occur regularly about what happens to  EMBED Equation.DSMT4  at x = 0. Most beginning calculus students (and a heck of a lot of teachers) say that f is neither increasing nor decreasing at x = 0 (some people use the word stationary). The function has a horizontal tangent line at x = 0 so its slope is neither positive nor negative so its neither increasing nor decreasing. Unfortunately, this isnt completely consistent with the most common definitions of increasing and decreasing. Definition: A function f is increasing on an interval if for any x1 and x2 on that interval, if x2 > x1 then  EMBED Equation.DSMT4 . A function f is decreasing on an interval if for any x1 and x2 on that interval, if x2 > x1 then  EMBED Equation.DSMT4 . In simple words, this says that a function is increasing on an interval if, for any two points in the interval, the point on the graph farther to the right is higher than the point on the graph to the left. Similarly a function is decreasing on an interval if, for any two points in the interval, the point on the graph farther to the right is lower than the point on the graph to the left. Clearly, for  EMBED Equation.DSMT4 , these definitions hold for the intervals (0, () and (((, 0). For any two positive x-values on the graph of y = x2, the one to the right will be higher (have a greater y-value) than the one to the left. For any two negative x-values, the one to the right will be lower than the one to the left. What many people find counterintuitive is that the definitions still hold if we change the intervals to [0, () and (((, 0]. For any two x-values on [0, (), the one to the right will be higher than the one to the left. This is true even if the left one is x = 0. Also, for any two x-values on (((, 0], the one to the right will be lower than the one to the left, even if the right one is x = 0. But this means that x = 0 is included in an interval where f is increasing and also included in an interval where f is decreasing. Students (understandably) hate that. The usual explanation of this seeming paradox is that the definitions of increasing and decreasing involve an interval of points and cant be applied to a single point. Thus, strictly by the definitions of increasing and decreasing, the question of whether f is increasing or decreasing at x = 0 makes no sense and so cant have a sensible answer. Unfortunately, the same reasoning should apply to x = 1 or x = (1 yet most people really want to say the f is increasing at x = 1 and decreasing at x = (1. (In fact, the AP sometimes asks questions about whether a function is increasing at a point. It is probably not a good idea to answer that the question doesnt make sense.) To solve this problem, people have come up with definitions for increasing (or decreasing) at a point that are consistent with the definitions above for intervals. Typically, they say that f is increasing at x = a if a is in the interior (not an endpoint) of an interval where f is increasing. Then we can say that  EMBED Equation.DSMT4  is increasing at x = 1 because it is increasing on the interval, say, (0.5, 1.5). In fact, we can say that f is increasing at any positive x-value x = c because c is on the interior of the interval  EMBED Equation.DSMT4  where f is increasing. By this definition, f is neither increasing nor decreasing at x = 0 because any interval that has x = 0 on the interior (not an endpoint) will include points where f is increasing and other points where f is decreasing. Now everybody is happy. So what does all this have to do with derivatives? There is a very important theorem that connects the derivative to intervals where a function is increasing or decreasing. Theorem: If f is differentiable on (a, b) and f ' > 0 for every x in (a, b), then f is increasing on (a, b). If f is differentiable on (a, b) and f ' < 0 for every x in (a, b), then f is decreasing on (a, b). For example, the function  EMBED Equation.DSMT4  has f ' > 0 for all x > 0 and so is increasing on (0, (). This is good enough for AP calculus. If the question was On what interval is the function  EMBED Equation.DSMT4  increasing? Justify your answer, the response f is increasing on (0, () because f ' > 0 on that interval would earn full credit. However, the theorem can be extended slightly to say that if f is continuous on [a, b] and differentiable on (a, b) and f ' > 0 for every x in (a, b), then f is increasing on [a, b]. Many college professors would prefer the answer to the above question to be f is increasing on [0, () because f ' > 0 on (0, (). On the AP, if the question were multiple choice, the choice given would be [0, (), not (0, (). (No, they would not offer both choices.) One final note. It should be clear that the converse of the theorem above, If f is continuous and increasing on an interval then f ' > 0 at all points in that interval, is not true. The function  EMBED Equation.DSMT4  is increasing on [0, () but its derivative is not positive at every point in that interval. A more interesting example is  EMBED Equation.DSMT4  which is increasing for all real x but has non-positive derivatives at an infinite number of points: f ' = 0 at x = ((, (3(, (5(, . . . f " 0 1 2 3 4 5 6 7 x y f f ' a x y y x -4 -3 -2 -1 0 1 2 3 4 f " f ' f " f ' x y x y y x y x y x y x y y x y ( x y x y x x  y x x y y x y x y x y x y x y x y x (   ( ( ( y x y x y x y x y y Graph of g' y x 15 10 5 y x  y H x y t H ' A t C Bt Graph of f ' t Graph of f " Graph of f ' x x x y Graph of f ' y x x  y    x x x  y y x y y x y x y x y x y x  x y y x  x y y x y x y x y x y x y x y x s t s t s t s t y x y x y x y x y x y x y x y x y x y x y x y x y x y x y x y x y x y x y x y x y x y x y x y x x y y x  x y Position (m) 120 100 80 60 40 20 0 x t  0 2 4 6 8 10 12 14 Time (s)   y x 4 3 2 1 -1 -2 -3 -4  -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 3 2 1 -1 -2 -3 -4 y x   y x  y x y x y x ( ( ( y x  5a. With an interest rate of 5%, the 10-year balance will be $1649. b. 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