Problems

Evaluate the following integrals.

In Problems 32-36, find the position y as a function of f given the velocity v = dy/dt and the value of y at one point of time.

In Problems 37-42, find the position y and velocity v as a function of t given the acceleration a and the values of y and v at f = 0 or t = 1.

37            a = t,                  v = 0 and y = 1 when t = 0

38            a =-32,            v =10 and y = 0 when t = 0

39             a = 3t²,                v = 1 and y = 2 when t = 0

40             a = 1 - √t,        v = -2 and y = 1 when t = 0

41              a = t^-3,               v = 1 and y = 0 when t = 1

42             a = -sin t,          v = 0 and y = 4 when r = 0

43             Which of the following definite integrals are undefined?

44              Find the function f such that f' is constant, f (0) = f'(0) and f (2) = f'(2).

45             An object moves with acceleration a = 6t. Find its position y as a function of f. given that y = 1 when t = 0 and y = 4 when t = 1.

46              Find the function h such that h" is constant, h(l) = 1, h(2) = 2, and h(3) = 3.

47              Suppose that F"(x) exists for all x. and let (x₀, y₀) and (x₁, y₁) be two given points. Prove that there is exactly one function G(x) such that

G(x₀) = y₀ G'(x₁) = y₁ G"(x) = F"(x) for all x.

48           Assume that F"(x) exists for all x, and let (x₁, y₁) and (x₂, y₂) be two points with x₁ x₂.

Prove that there is exactly one function G(x) such that G"(x) = F"(x) for all x, and the graph of G passes through the two points (x₁, y₁) and (x₂, y₂).