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AP Calculus 5.4 Worksheet Day 1 All work must be shown in this course for full credit. Unsupported answers may receive NO credit. For questions 1 10, use the Fundamental Theorem of Calculus (Evaluation Part) to evaluate each definite integral. Use your memory of derivative rules and/or the chart from your notes. You should start making a list of all the rules on ONE page! 4 5 x3 1. 5 x 2 dx 2 3 3 2 2 3 1 3. 1 1 x 2 1 4. dx 1 2 1 x2 dx 12 x 5 dx 5. dx x 2. 7x dx 6. 0 5 5 6 dx 7. 8. 2 3 4 sec2 x dx 9. sin 5x dx 2 1 0 Need more practice e 2 x dx 10. age 303 #27 40 For questions 11 and 12, setup and evaluate an expression involving definite integrals in order to find the total AREA of the region between the curve and the x-axis. [No Calculator!] 11. y 3x 2 3 on the interval 2 < x < 2 12. y x on the interval 0 < x < 9 For questions 13 16, find the average value of the function on the specified interval without a calculator. 13. g x 15. y 9 3 3xx 2 on the interval [0, 4] 5x iif 0 x 2 12 x if 2 x 1 12 14. h x csc x ccot x on the interval 16. f x sec2 x on the interval 0, 4 2 ,4 17. Including start-up costs, it costs a printer $50 to print 25 copies of a newsletter, after which the marginal cost (in 2 dollars per copy) at x copies is given by C ' x . Find the total cost of printing 2500 newsletters. x x 18. Suppose f x x 2 3x 1 . Find K so that x f t dt a f t d dt if a = 1 and b = 2. K b 9 f ' x dx 19. If you know 15 1 5 , and you know f 7 4 , what does f 9 ? 7 20. The graph of h ' x is given below. If h ( 2) = 6, what does h (3) = ? x 21. The graph of B ' x is given below. If you know that B (0) = 5, what does B (5) = ? x AP Calculus 5.4 Worksheet Day 2 x f t dt . If you know the graph of f (t) given below has odd symmetry and is periodic (with period = 2), 1. Let F x 0 1 and f t dt 0 f (t) 4 , complete the following table: 3 x 1 0 1 2 3 F(x) t 1 x ln 2 2. [Calculator required] If f x sin t d dt , and f (3) = 4, what does f (5) = ? a x 3. If a is a constant and g x w t dt , what is g ' x ? _____________________ w t dt , what is g ' x ? _____________________ a a 4. If a is a constant and g x x 5. Find 1 e5t5t dt . 5 t dt , find y ' . 0 2 ssin u du . Find k ' x . 3 ccos u d 8. Find dx x 9. What is the linearization of f x t3 6. If y 3 x 7. k x x x d dx cos3 t d dt at x ? 7 2 p4 x p 1 dp 2 x 10. Let H x f t dt , where f is the continuous function with domain [0, 12] shown below. 0 Graph of f (x) a) Find H (0) b) Is H (12) positive or negative? Explain. c) Find H ' x and use it to evaluate H ' 0 . d) When is H (x) increasing? Justify your answer. e) Find H '' x . f) When is H (x) concave up? Justify your answer. g) At what x-value does H (x) achieve its maximum value? Justify your answer. 12. The graph of a differentiable function f on the interval [ 2, 10] is shown in the figure below. The graph of f has a horizontal tangent line at x = 4. x Let h x f t dt for 9 Graph of f 2 < x < 10. 4 3 a) Find h (4), h ' 4 , and h '' 4 2 1 b) On what intervals is h increasing? Justify your answer. 2 2 4 1 c) On what intervals is h concave downward? Justify your answer. 10 f x dx using 6 subintervals of length = 2. d) Find the Trapezoidal Sum to approximate 2 6 8 10 p x w t dt , what is g ' x ? 13. If q (x) and p (x) are differential functions of x and g x q x 14. Find d dx sin x 1 t 3 dt 15. Find 1 10 16. If y u 2 du , find y ' . ln 2 17. Find 3xx 2 3 d dx d dx x3 coss 2t d 2t dt x2 x3 2 et dt d sin x x2 3 x 2 et d dt . At what value of x is f (x) a minimum? 18. Let f x 2 A B C D E none 0.5 1.5 2 3 19. The function g is define and differentiable on the closed interval [ 7, 5] and satisfies g (0) = 5. The graph of g ' x , the derivative of g, consists of a semicircle and three line segments as shown in the figure. Graph of g (x) a) Write an expression for g (x). b) Use your expression to find g (3) and g ( 2). c) Find the x-coordinate of each point of inflection of the graph of g (x) on the interval ( 7, 5). Explain your reasoning. t 20. Let s t f x dx be the position of a particle at time t (in seconds) as the particle moves along the x-axis. 0 The graph of the differentiable function f is shown below. Use the graph to answer the following questions. ! " # ! $ % & ! " ' ( $ ) # * $ ( + ' " ! , ! ! " - $ t = 4? Justify your answer. (7, 6.5) (6, 6) b) Is the acceleration of the particle at time t = 4 positive or negative? Justify your answer. c) Is the particle speeding up or slowing down at time t = 4? Explain. d) When does the particle pass through the origin? Explain. e) Approximately when is the acceleration zero? f) When is the particle moving toward the origin? Away from the origin? g) On which side of the origin does the particle lie at time t = 9? x 21. Suppose x f t dt 1 x2 2 x 1 . Find f (x). 22. Find g (4) if g t dt 0 x co cos x .