Practice Exercises

A1. 

A2. 

A3. 

(A) –2
(B) 4
(C) –1
(D) 2 

A4. 

A5. 

A6. 

(A) 1
(B) 
(C) –1
(D) 2

A7. 

A8. 

A9. 

A10. 

A11. 

A12. 

A13. 

(A) –ln 2
(B) 
(C) 
(D) ln 2

A14. If we let x = 2 sin θ, then  is equivalent to

A15. 

(A) 
(B) 1
(C) 
(D) 0

A16. 

*A17. 

(A) –1
(B) e + 1
(C) 1
(D) e – 1

A18. 

A19. 

A20. 

A21. 

A22. 

(A) e
(B) 2 + e
(C) 
(D) 1 + e

A23. 

(A) ln 2
(B) 1 + e
(C) –ln 2
(D) 

A24. If we let x = tan θ, then  is equivalent to

A25. If the substitution  is used, then  is equivalent to

A26. The table above shows some values of continuous function f and its first derivative. Evaluate .

(A) –1/2
(B) 3
(C) 4
(D) –1

A27. Using a midpoint Riemann Sum with 3 equal width subintervals, find the approximate area of the shaded region above.

(A) 19
(B) 36
(C) 38
(D) 54

A28. The graph of a continuous function f passes through the points (4,2), (6,6), (7,5), and (10,8). Using trapezoids, we estimate that 

(A) 30
(B) 32
(C) 33
(D) 41

A29. The area of the shaded region in the figure above is equal to ln 3. If we approximate ln 3 using both a left Riemann Sum and a right Riemann Sum with 2 equal width subintervals as shown, which inequality follows?

A30. Let . We estimate A using the L, R, and T approximations with n = 100 subintervals. Which is true?

(A) L < A < T < R
(B) L < T < A < R
(C) R < A < T < L
(D) R < T < A < L

A31. 

A32. 

A33. The average value of  on its domain is

(A) 2
(B) 4
(C) 2π
(D) 4π

A34. The average value of cos x over the interval  is

A35. The average value of csc²x over the interval from  to  is

A36. Choose the Riemann Sum whose limit is the integral .

A37. Choose the Riemann Sum whose limit is the integral .

A38. Choose the integral that is the limit of the Riemann Sum

A39.  is equal to

B1. Find the average value of function f, as shown in the graph below, on the interval [0,5].

(A) 4
(B) 5
(C) 7
(D) 8

B2. The integral  gives the area of

(A) a circle of radius 4
(B) a semicircle of radius 4
(C) a quadrant of a circle of radius 4
(D) half of an ellipse

Use the graph of function f, shown below, for Questions B3–B6.

B3. Calculate the average value of f(x) on the interval [0,6]. Which of the following intervals contains x = c, where f(c) is the average value of f on [0,6]?

I. [0,2]
II. [2,4]
III. [4,6]

(A) I only
(B) II only
(C) III only
(D) I and II only

B4. 

B5. Let ; then g′(1) =

(A) 3
(B) 4
(C) 6
(D) 8

B6. Let h(x) = x² – f(x). Find .

(A) 38
(B) 58
(C) 70
(D) 74

B7. If f(x) is continuous on the closed interval [a,b], then there exists at least one number c, a < c < b, such that  is equal to

(A) 
(B) f′(c)(b – a)
(C) f(c)(b – a)
(D) 

B8. If f(x) is continuous on the closed interval [a,b] and k is a constant, then  is equal to

(A) k(b – a)
(B) k[f(b) – f(a)]
(C) 
(D) 

B9. 

B10. If , then F′(u) is equal to

(A) –6u (2 – u²)²
(B) (2 – u²)³ – 1
(C) (2 – u²)³
(D) –2u (2 – u²)³

B11. 

B12. If x = 4 cos θ and y = 3 sin θ, then  is equivalent to 

B13. A continuous function f takes on the values shown in the table below. Estimate  using a left rectangular approximation with five subintervals.

(A) 144
(B) 170
(C) 186
(D) 196

B14. Find the value of x at which the function y = x² reaches its average value on the interval [0,10].

(A) 5.313
(B) 5.774
(C) 6.083
(D) 18.257

B15. The average value of  on the interval 0 ≤ x ≤ 5 is

(A) 6.25
(B) 6.41
(C) 9.2
(D) 10