AP Calculus BC Unit 6 Practice Test & Exam | 37 Questions

1. Oil is leaking from a storage tank at a rate modeled by the function $R(t)$, where $R(t)$ is measured in gallons per hour and $t$ is measured in hours since the leak began. Which of the following expressions represents the total amount of oil, in gallons, that has leaked from the tank during the first 3 hours?

(A)$R(3) - R(0)$(B)$\frac{1}{3} \int_{0}^{3} R(t) \, dt$(C)$\int_{0}^{3} R(t) \, dt$(D)$R'(3)$

2. Which of the following is equal to $\int \frac{3x^2+2x}{x+1} \, dx$ ?

(A)3x - 1 + $\ln$|x+1| + C(B)$\frac{3}{2}$x^2 - x + $\ln$|x+1| + C(C)$\frac{3}{2}$x^2 + x - $\ln$|x+1| + C(D)3x^2 - x + $\frac{1}{(x+1)^2} $+ C

3. Find the value of $\int \frac{1}{x^2+6x+13} \, dx$.

(A)$\ln$|x^2+6x+13| + C(B)$\arctan\left$($\frac{x+3}{2}\right$) + C(C)$\frac{1}{2}\arctan\left$($\frac{x+3}{2}\right$) + C(D)$\frac{1}{2}\ln\left$|$\frac{x+3}{2}\right$| + C

4. Evaluate $\int_{1}^{1+\sqrt{3}} \frac{1}{x^2-2x+2} \, dx$.

(A)$\frac{\pi}{6}$(B)$\frac{\pi}{4}$(C)$\frac{\pi}{3}$(D)$\frac{\pi}{2}$

5. Which of the following is an antiderivative of $f(x) = x \cos(2x)$?

(A)$\frac{1}{2}x \sin(2x) - \frac{1}{4} \cos(2x) + C$(B)$\frac{1}{2}x \sin(2x) + \frac{1}{4} \cos(2x) + C$(C)$x \sin(2x) + \cos(2x) + C$(D)$2x \sin(2x) + \cos(2x) + C$

6. Selected values for a twice-differentiable function $f$ and its derivative $f'$ are given in the table below. | $x$ | $f(x)$ | $f'(x)$ | |---|---|---| | 0 | 3 | 1 | | 2 | 5 | -2 | What is the value of $\int_0^2 x f''(x) \, dx$?

(A)-6(B)-2(C)2(D)6

7. What is the value of $\int_1^e x \ln(x) \, dx$?

(A)$\frac{e^2 - 1}{4}$(B)$\frac{e^2 + 1}{4}$(C)$\frac{e^2 - 1}{2}$(D)$\frac{e^2 + 1}{2}$

8. Which of the following is the antiderivative of $f(x) = \frac{4x+1}{x^2-3x-10}$?

(A)$3\ln|x-5| + \ln|x+2| + C$(B)$\ln|x-5| + 3\ln|x+2| + C$(C)$2\ln|x^2-3x-10| + C$(D)$3\ln|x-5| - \ln|x+2| + C$

9. Which of the following expressions is equivalent to the integrand $\frac{x^2+2x-5}{x^2-x-2}$ and can be used to evaluate $\int \frac{x^2+2x-5}{x^2-x-2} \, dx$?

(A)$1 + \frac{1}{x-2} + \frac{2}{x+1}$(B)$1 + \frac{2}{x-2} + \frac{1}{x+1}$(C)$\frac{1}{x-2} + \frac{2}{x+1}$(D)$1 - \frac{1}{x-2} + \frac{2}{x+1}$

10. Evaluate $\int_{2}^{3} \frac{1}{x^2-1} \, dx$.

(A)$\frac{1}{2} \ln \frac{3}{2}$(B)$\ln \frac{3}{2}$(C)$\frac{1}{2} \ln 3$(D)$\frac{1}{2} \ln \frac{4}{3}$

11. Evaluate the improper integral $\int_{0}^{\infty} 2x e^{-x} \, dx$.

(A)0(B)1(C)2(D)The integral diverges

12. Which of the following statements is true regarding the integral $\int_{-2}^{1} \frac{1}{x^2} \, dx$?

(A)The integral converges to -1.5.(B)The integral converges to 1.5.(C)The integral converges to 3.(D)The integral diverges.

13. Evaluate $\int_{1}^{\infty} \frac{4}{1+x^2} \, dx$.

(A)$\pi$(B)$\frac{\pi}{2}$(C)2$\pi$(D)The integral diverges

14. Consider the following three integrals: I. $\int 4x e^{x^2} \, dx$ II. $\int 4x e^x \, dx$ III. $\int \frac{4}{x^2 - 5x + 6} \, dx$ Which of the following correctly matches each integral with the most appropriate technique for finding its antiderivative?

(A)I: Integration by Parts, II: Substitution, III: Partial Fractions(B)I: Substitution, II: Integration by Parts, III: Partial Fractions(C)I: Substitution, II: Integration by Parts, III: Substitution(D)I: Integration by Parts, II: Substitution, III: Completing the Square

15. To evaluate the integral $\int x^2 \ln(x) \, dx$ using integration by parts, which of the following choices for $u$ and $dv$ is most appropriate?

(A)$u = x^2$ and $dv = \ln(x) \, dx$(B)$u = \ln(x)$ and $dv = x^2 \, dx$(C)$u = x^2 \ln(x)$ and $dv = dx$(D)$u = \frac{1}{x}$ and $dv = x^3 \, dx$

16. For which of the following integrals is the method of partial fraction decomposition the most appropriate technique to select?

(A)$\int \frac{2x}{x^2 - 9} \, dx$(B)$\int \frac{3}{x^2 + 9} \, dx$(C)$\int \frac{5}{x^2 - 4x - 12} \, dx$(D)$\int \frac{4}{(x-3)^2} \, dx$

17. The function $f$ is continuous on the closed interval $[0, 6]$ and has values that are given in the table below. | $x$ | 0 | 2 | 5 | 6 | | :--- | :--- | :--- | :--- | :--- | | $f(x)$ | 5 | 9 | 7 | 3 | Using the subintervals $[0, 2]$, $[2, 5]$, and $[5, 6]$, what is the trapezoidal approximation of $\int_{0}^{6} f(x) \, dx$?

(A)36(B)42(C)43(D)44

18. Let $f$ be a function that is continuous, strictly decreasing, and concave up on the closed interval $[a, b]$. Let $L_n$, $R_n$, and $T_n$ denote the approximations of $\int_{a}^{b} f(x) \, dx$ using $n$ equal subintervals for a Left Riemann sum, a Right Riemann sum, and a Trapezoidal sum, respectively. Which of the following correctly orders these approximations?

(A)$R_n < T_n < L_n$(B)$T_n < R_n < L_n$(C)$L_n < T_n < R_n$(D)$T_n < L_n < R_n$

19. Let $f$ be the function given by $f(x) = x^2 - 2x + 4$. What is the approximation for $\int_{0}^{6} f(x) \, dx$ using a Midpoint Riemann sum with 3 subintervals of equal length?

(A)40(B)58(C)64(D)88

20. Which of the following definite integrals is equivalent to the limit given by $\displaystyle \lim_{n \to \infty} \sum_{k=1}^{n} \sqrt{3 + \frac{4k}{n}} \cdot \frac{4}{n}$ ?

(A)$\displaystyle \int_{3}^{7} \sqrt{x} \, dx$(B)$\displaystyle \int_{0}^{4} \sqrt{x} \, dx$(C)$\displaystyle \int_{3}^{7} \sqrt{3+x} \, dx$(D)$\displaystyle \int_{0}^{3} \sqrt{3+x} \, dx$

21. Which of the following limits is equal to the definite integral $\displaystyle \int_{2}^{5} (x^2 + 1) \, dx$ ?

(A)$\displaystyle \lim_{n \to \infty} \sum_{k=1}^{n} \left[ \left( \frac{3k}{n} \right)^2 + 1 \right] \frac{3}{n}$(B)$\displaystyle \lim_{n \to \infty} \sum_{k=1}^{n} \left[ \left( 2 + \frac{3k}{n} \right)^2 + 1 \right] \frac{3}{n}$(C)$\displaystyle \lim_{n \to \infty} \sum_{k=1}^{n} \left[ \left( 2 + \frac{3k}{n} \right)^2 + 1 \right] \frac{1}{n}$(D)$\displaystyle \lim_{n \to \infty} \sum_{k=1}^{n} \left[ \left( 2 + \frac{5k}{n} \right)^2 + 1 \right] \frac{5}{n}$

22. Let $h$ be the function defined by $h(x) = \int_{1}^{2x} \sqrt{t^3 + 1} \, dt$. What is the value of $h'(1)$ ?

(A)3(B)6(C)9(D)18

Refer to the figure below.

23. The graph of the function $f$, shown above, consists of three line segments defined on the interval $[-4, 4]$. Let $g$ be the function defined by $g(x) = \int_{-4}^{x} f(t) \, dt$. On which of the following open intervals is the graph of $g$ both decreasing and concave down?

(A)(-4, -2)(B)(-2, 0)(C)(0, 2)(D)(2, 4)

Refer to the figure below.

24. The graph of the continuous function f is shown above for 0 $\le $x $\le $8 . Let h be the function defined by h(x) = $\int_{0}^{x} $f(t) $\, $dt . The graph of f consists of two semicircles: one above the x-axis with radius 2 centered at x=2 , and one below the x-axis with radius 2 centered at x=6 . Which of the following correctly orders h(2) , h(4) , and h(8) from least to greatest?

(A) h(8) < h(2) < h(4) (B) h(8) < h(4) < h(2) (C) h(2) < h(8) < h(4) (D) h(4) < h(2) < h(8)

25. If $\int_{-2}^{6} $h(x) $\, $dx = 12 and $\int_{-2}^{2} $h(x) $\, $dx = 5, what is the value of $\int_{6}^{2} $(2h(x) - 3) $\, $dx?

(A)-26(B)-17(C)-2(D)2

26. The graph of the function f, defined on the closed interval [0, 6], consists of a semicircle of radius 2 centered at (2,0) and a line segment from (4,0) to (6,4). What is the value of $\int_{0}^{6} $(3 - f(x)) $\, $dx?

(A)14 - 2$\pi$(B)14 + 2$\pi$(C)22 - 2$\pi$(D)18 - 4$\pi$

27. Evaluate the definite integral $\int_{e}^{e^4} \frac{1}{x\sqrt{\ln x}} \, dx$.

(A)2(B)4(C)2$\sqrt{3}$(D)e^2 - $\sqrt{e}$

28. Let $f$ be the function defined by $f(x) = \int_{0}^{x^2} \sqrt{3t + 4} \, dt$. What is the value of $f'(2)$?

(A)4(B)8(C)16(D)20

29. Which of the following is equal to $\int \frac{3x^2 - 5x + 2}{\sqrt{x}} \, dx$ ?

(A)$\frac{6}{5}x^{5/2} - \frac{10}{3}x^{3/2} + 4x^{1/2} + C$(B)$\frac{3x^3 - 5x^2 + 2x}{\frac{2}{3}x^{3/2}} + C$(C)$6x^{5/2} - 10x^{3/2} + 4x^{1/2} + C$(D)$\frac{3}{2}x^{3/2} - \frac{5}{2}x^{1/2} - x^{-3/2} + C$

30. Find the indefinite integral $\int \left( 4^x + \csc x \cot x - e^2 \right) \, dx$.

(A)$\frac{4^x}{\ln 4} - \csc x - e^2 x + C$(B)$4^x \ln 4 - \csc x - \frac{e^3}{3} + C$(C)$\frac{4^x}{\ln 4} + \csc x - e^2 x + C$(D)$\frac{4^{x+1}}{x+1} - \cot x - 2e x + C$

31. Evaluate the indefinite integral $\int \frac{6x^2}{\sqrt{2x^3 + 5}} \, dx$.

(A)2$\sqrt{2x^3 + 5} $+ C(B)6$\sqrt{2x^3 + 5} $+ C(C)$\frac{1}{2}\sqrt{2x^3 + 5} $+ C(D)3(2x^3 + 5)^{-1/2} + C

32. Which of the following is equivalent to the definite integral $\int_{0}^{\pi/4} \tan(x) \sec^2(x) \, dx$?

(A)$\int_{0}^{\pi/4} $u $\, $du(B)$\int_{0}^{1} $u $\, $du(C)$\int_{0}^{1} $u^2 $\, $du(D)$\int_{0}^{\sqrt{2}/2} $u $\, $du

33. Water is pumped into a tank at a rate modeled by the function $R(t)$, measured in gallons per minute, where $t$ is the time in minutes for $0 \le t \le 10$. The graph of $R(t)$ is shown in the figure above and consists of three line segments connecting the points $(0, 0)$, $(4, 20)$, $(8, 20)$, and $(10, 0)$. Which of the following gives the total amount of water, in gallons, pumped into the tank from time $t=0$ to $t=10$?

(A)80(B)100(C)140(D)200

34. [Skill: 3.D | Topic: 6.1] Water flows into a storage tank at a rate modeled by the function $R(t)$, measured in gallons per hour, where $t$ is the time in hours for $0 \le t le 8$. The graph of $R(t)$ is shown below. [Image Cue]: A piecewise linear graph on the interval $[0, 8]$ with the vertical axis labeled "Rate (gallons per hour)" and the horizontal axis labeled "Time (hours)". The graph consists of three line segments connecting the points $(0, 0)$, $(2, 100)$, $(5, 100)$, and $(8, 0)$. Which of the following is the best interpretation of the total amount of water, in gallons, that flowed into the tank from $t = 0$ to $t = 8$ hours?

(A)400(B)550(C)800(D)1100

AP Calculus BC Unit 6: Integration and Accumulation of Change Practice Exam

Exam Details

A. Multiple-Choice Questions (34 Questions)

B. Short Answer Questions (3 Questions)

AP Calculus BC Unit 6: Integration and Accumulation of Change Practice Exam

Exam Details

A. Multiple-Choice Questions (34 Questions)

B. Short Answer Questions (3 Questions)