Chapter 5 Review. 1. [No Calculator] Evaluate using the FTOC (the evaluation part) 2. [No Calculator] Evaluate using geometry

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1 AP Calculus Chapter Review Name: Block:. [No Calculator] Evaluate using the FTOC (the evaluation part) a) d b) d. [No Calculator] Evaluate using geometry a) d c) 6 8 d. [No Calculator] Evaluate each derivative. d sec a) tant 9 dt d b) Find h' if h 4t 9 dt. 4 d tan dt 9 d d) Find h' if h dt. 8 6 c) lnt 9 4. [No Calculator] Given the graph of f () as shown and the definition of g () = f t dt a) Find g ( ), g ', g '' (, ) b) Over what interval is g () increasing. Show your work and eplain your reasoning. (, ) c) Over what interval is g () concave up? Show your work and eplain your reasoning. Graph of f d) Graph g ()

2 . [No Calculator] The graph of the function f shown below consists of three line segments. a) Let g be the function given by g For each of g ( ), g ', and '' state that it does not eist. f t dt. 4 g, find the value or b) For the function g defined in part a, find the -coordinate of each point of inflection of the graph of g on the open interval 4 < <. Eplain your reasoning. c) Let h be the function given by h f t dt. Find all values of in the closed interval 4 < < for which h () =. d) For the function h defined in part c, find all intervals on which h is decreasing. Eplain your reasoning. 6. [Calculator] The temperature, in degrees Celsius ( C), of the water in a pond is a differentiable function W of time t. The table below shows the water temperature as recorded every days over a -day period. a) Use data from the table to find an approimation for W '. Show the computations that lead to your answer. Indicate units of measure. b) Approimate the average temperature, in degrees Celsius, of the water over the time interval < t < days by using a trapezoidal approimation with subintervals of length t = days. t W (t) (days) ( C) / c) A student proposes the function P, given by Pt te t at time t, where t is measured in days and P (t) is measured in degrees Celsius. Find P '. Using appropriate units, eplain the meaning of your answer in terms of water temperature., as a model for the temperature of the water in the pond d) Use the function P defined in part c to find the average value, in C, of P (t) over the time interval < t < days.

3 7. [Calculator] For < t <, the rate of change of the number of mosquitoes on Tropical Island at time t days is modeled by Rt tcos t mosquitoes per day. There are mosquitoes on Tropical Island at time t =. a) Show that the number of mosquitoes is increasing at time t = 6. b) At time t = 6, is the number of mosquitoes increasing at an increasing rate, or is the number of mosquitoes increasing at a decreasing rate? Give a reason for your answer. c) According to the model, how many mosquitoes will be on the island at time t =? Round your answer to the nearest whole number. d) To the nearest whole number, what is the maimum number of mosquitoes for < t <? Show the analysis that leads to your conclusion. 8. [No Calculator] Suppose f d, f d, and g d 7. Find each of the following: a) g d b) f d c) g f d d) f e) f g d f) 4 g d

4 9. [No Calculator] Suppose H ln( t ) dt for the interval [, ]. a) Use MRAM to approimate H () using 4 equal subdivisions. b) When is H () decreasing? Justify your response. c) If the average rate of change of H () on [, ] is k, what is the value of ln( t ) dt in terms of k.. [No Calculator] Let H a) Find H () f t dt, where f is the continuous function with domain [, ] shown below. Graph of f () b) Is H () positive or negative? Eplain. c) Find H ' and use it to evaluate ' H. d) When is H () increasing? Justify your answer. e) Find H ''. f) When is H () concave up? Justify your answer. g) At what -value does H () achieve its maimum value? Justify your answer.

5 b. [No Calculator] If f d a b, then A a + b + a b a f d B C D E b a 4a b b a b a. [No Calculator] Let t f e dt. At what value of is f () a minimum? A none B. C. D E. [Calculator] If ln sin f t dt, and f () = 4, what does f () =? a

6 Time (in seconds) v (t) (in ft/sec) Velocity (in feet per second) Time (in seconds) 4. The graph of the velocity v (t), in ft/sec, of a car traveling on a straight road, for < t <, is shown above. A table of values for v (t), at second intervals of time t, is shown to the right of the graph. a) During what intervals of time is the acceleration of the car positive? Give a reason for your answer. b) Find the average acceleration of the car, in ft/sec, over the interval < t <. c) Approimate correct units, eplain the meaning of this integral. v t dt with a Riemann sum, using the midpoints of five subintervals of equal length. Using