Extra Problems

1            Evaluate

2            Evaluate

3            Evaluate

4            Evaluate

5            If F'(x) = l/(2x - 1)² for all x 1/2, find F(2) - F(l).

6             If G'(t) =for all t > - 1/4, find G(2) - G(0).

7            A particle moves with velocity . How far does it move from times t₀ = 1 to t₁ = 5?

8            A particle moves with velocity . How far does it move from times t₀ = 1 to t₁ = 4?

9            A particle moves with velocity v = (t + l)(2t + 3). If it has position y₀ = 0 at time t = 0, find its position at time t = 10.

10            A particle moves with acceleration a = 1/t⁴. If it has velocity v₀ = 4 and position y₀ = 2 at time t = 1, find its position at time t = 3.

11             Find the area of the region under the curve y = 1/√x, 1 ≤ x ≤ 4.

12            Find the area of the region under the curve y = √x - x√x, 0 ≤ x ≤ 1. In Problems 13-30, evaluate the integral.

13            14

15            16

17            18

19            20

21            22

23            24

25            26

27            28

29            30

31           Differentiate

32           Differentiate

33           Differentiate

34           Differentiate

35            Find the function F such that F'(x) = x - 1 for all x, and the minimum value of F(x) is b..

36            Find the function F such that F"(x) = x for all x, F(0) = 1, and F(l) = 1.

37            Find the function F such that F"(x) = 6 for all x, F(x) has a minimum at x = 1, and the minimum value is 2.

38            Find all functions F such that F"(x) = 1 + x^-3 for all positive x.

39            Find the function F such that and F(0) = 1.

40            Find the value of b such that the area of the region under the curve y = x{b - x), 0 ≤ x ≤ b, is 1.

41            Suppose f is increasing for a ≤ x ≤ b, and Δx = (b - a)/n where n is a positive integer. Show that

42            Suppose f is continuous for a ≤ x ≤ b. Show that

43             Find the area of the top half of the ellipse x²/a² + y²/b² = 1 using the formula

44            Evaluate using the formula

45            Find

46            Suppose f(t) is continuous for all t and let Prove that G"(x) = f(x).

47            Prove that for any continuous functions f and g,

48            Prove Schwartz' Inequality, Hint: Use the preceding problem.

49            Suppose f is continuous and dx is positive infinitesimal. Show that Hint: For each positive real c, Use this to show that

50            Suppose f is continuous, n is an integer, and dx is positive infinitesimal. Prove that