Please provide explanations for 43 and 45

numbers a and b is the number (a + b)/2 36. a. y' = sec2 0 b. y' = Ve c. y' = VO - sec2 0 c in the conclusion of the Mean Value The any interval [a, b ] is c = (a + b)/2. In Exercises 37-40, find the function with the given derivative whose T 57. Graph the function graph passes through the point P. f(x) = sin x sin (x + 2) - sin? ( 37. f'(x) = 2x - 1, P(0, 0) What does the graph do? Why does the func 38. 8' (x) = -+ 2x, P (-1, 1) Give reasons for your answers. 58. Rolle's Theorem 39. r'(0) = 8 - csc2 0, P( #, 0) a. Construct a polynomial f(x) that has zero 1, and 2. 40. r'(t) = sec t tan t - 1, P(0, 0) b. Graph f and its derivative f' together. How related to Rolle's Theorem? Finding Position from Velocity or Acceleration Exercises 41-44 give the velocity v = ds/ di and initial position of an c. Do g(x) = sin x and its derivative g' illustra object moving along a coordinate line. Find the object's position at phenomenon as f and f'? time t. 59. Unique solution Assume that f is continuou 41. v = 9.8t + 5, s(0) = 10 42. v = 32t - 2, s(0.5) = 4 differentiable on (a, b). Also assume that f(a) a posite signs and that f' # 0 between a and b. Sho 43. v = sin Trt, s(0) = 0 44. v = 21, S(TT 2 ) = 1 COST, exactly once between a and b. 60. Parallel tangents Assume that f and g are d Exercises 45-48 give the acceleration a = d's/ di2, initial velocity, [a, b ] and that f(a) = g(a) and f(b) = g(b). She at least one point between a and b where the tangen and initial position of an object moving on a coordinate line. Find the of f and g are parallel or the same line. Illustrate wi object's position at time t. dinates 45. a = 32, v(0) = 20, s(0) = 5 61. Suppose that f'(x) = 1 for 1 S x $ 4. Show f(1)