4.3 EXERCISES 1. Explain exactly what is meant by the statement that 5-6 Sketch the area represented by g(x). Then find g'(x) in two 'differentiation and integration are inverse processes." ways: (a) by using Part 1 of the Fundamental Theorem and (b) by 2, Let g(x) = fif(1) di, where f is the function whose graph is evaluating the integral using Part 2 and then differentiating. shown. 6. 9(x) = S." (2 + sin 1) dt (a) Evaluate g(x) for x = 0, 1, 2, 3, 4, 5, and 6. 5. g ( x ) = $ 12 di (b) Estimate g(7). (c) Where does g have a maximum value? Where does it have a minimum value? 7-18 Use Part 1 of the Fundamental Theorem of Calculus to find (d) Sketch a rough graph of g. the derivative of the function. 7. g ( x ) = [" Vi+ + de (3 9(x) = ["cos(12 ) at 9. g(s) = [(1-12)8 di 10 . h ( u ) = Jo 7 + I de 11 . F (x ) = [ VI + sec i dt 3. Let g(x) = So f(t) dt, where f is the function whose graph is shown. Hint : [ VI + sec i dt = - [" VI + sect dt (a) Evaluate g(0), g(1), g(2), g(3), and g(6). (b) On what interval is g increasing? (c) Where does g have a maximum value? 12. R(y) = 13 sin t dt d) Sketch a rough graph of g. 13. h(x) = sin't dt 14. h(x) = + 1 15. y = J, 1+ 13 dt 16 . y = cos 20 do 17. y = - 0 tan 0 do 18 . y = VI+ 1 2 dt 19-38 Evaluate the integral. 4. Let g(x) = Sof(t) dt, where f is the function whose graph is shown. 19 . ( x 2 + 2 x - 4 ) dx 20 . x 100 dx (a) Evaluate g(0) and g(6). (b) Estimate g(x) for x = 1, 2, 3, 4, and 5. (c) On what interval is g increasing? 27 . 2 (3 + 3 - 4+2 + 3+ ) dt (22 3 5 (1 - 803 + 160 " ) do (d) Where does g have a maximum value? (e) Sketch a rough graph of g. 23. Vx dx 24. x - 2/3 dx (f) Use the graph in part (e) to sketch the graph of g'(x). Compare with the graph of f. 25 . " sin 0 de 26 . T dx -2 27. (u + 2 ) ( 1 - 3 ) du 28 . [ # ( 4 - 1 ) Vide 0 29 (4 2 + x2 Vx 30 . ( 3u - 2 ) ( u + 1 ) du 31. 7/ 2 J T. /6 csc t cot t dt 32. 7/3 csc20 do328 CHAPTER 4 Integrals 33. (1 + r) dr 34 . ( 3 5 + 1 ds 54.) g(x) - [ Isint di 56. 9(x) = Jinx 2 + 15 di 55. h (x) = [ cos(1?) di 35. (205 + 30 36. (" 3 dz 57. Let F(x) = [ cos ' di. Find an equation of the tangent line 37. ["f(x) dx where f(x) = sin x if 0 5 x