d 25. dx J d 27. 4. Let G(x) = sint dt. Which of the following stateme dt correct? 29. Let A (a) G is the composite function sin(x3). (a) Calc (b) G is the composite function A(x), where (b) Find satisfying A (x ) = sint dt 12 dt (c) G(x) is too complicated to differentiate. (d) The Product Rule is used to differentiate G. (e) The Chain Rule is used to differentiate G. plain. (f) G'(x) = 3x2 sin(x3). 30. Ma ower limit 6 . f (x ) = 1 - x+cosx, a=0 function and show 7. f (x) = e2x, a = 0 8. f (x) = e-*, a=-1 In Exercises 9-12, compute or approximate the corresponding! values and derivative values for the given area function. In some approximations will need to be done via a Riemann sum. 9 . F ( x ) = V12 + t dt . Find F ( 0 ) , F ( 3 ) , F ' ( 0 ) , and F (8 y = f (x ) X 10 -8 10 -8 (B) FIGURE 10 te the derivative. 1/x 34. cos' t dt dx X1 36. Vt dt dx . (x) = A(x4) - A(x2). d 3u 38. Vx2 + 1dx du -u th f(x) as in Figure 11, let B(x) = f(t) dt f(t) dt and I max of A on [0, 6]. d max of B on [0, 6]. B(x) valid on [2, 4](0), and F'(2) s where 12. T() = 1 tan 0 de. Find T(0), T (x /3), T'(0), and T'(7 /3). + . This Exercises 13-22, find formulas for the functions represented integrals. u *du 14. ( 121 2 - 81 ) dt 1.3. 16. sinu du sec2 0 de 15. -x /4 Jo 3u du 18. erdt 17. X x/4 20. t dt sec u du 19. x/2 X 9x +2 dt e-" du 22. va- 21. J3x 23. Verify It| dt = =x|x|. Hint: Consider x 2 0 and re rately. 24. Verify It' dt = =xx\\'. Hint: Consider x 2 0 o separately. In Exercises 25-28, calculate the derivative. cot 25. de dx [(1 - 913 ) dt 26. d d ftP R S y = f (x) FIGURE 12 Graph of y = f(x). 44. Let A(x) = f (1) dt, with f (x) as in Figure 12. (a) Where does A have its absolute maximum over the interva (b) Where does A have its absolute minimum over the interva (c) On what interval is A increasing? In Exercises 45-46, let A(x) = f (t ) dt . 45 Area Functions and Concavity Explain why t tatements are true. Assume f is differentiable. (a) If A has an inflection point at x = c, then f' (c) = 0. (b) A is concave up if f is increasing. (c) A is concave down if f is decreasing. 46. Match the property of A with the corresponding propert of f. Assume f is differentiable. Area function A (a) A is decreasing. (b) A has a local maximum. (c) A is concave up. (d) A goes from concave up to concave down