AP Calculus BC Unit 9: Parametric Equations, Polar Coordinates, and Vector-Valued Functions (BC ONLY) Practice Exam

Assessment for Unit 9: Parametric Equations, Polar Coordinates, and Vector-Valued Functions (BC ONLY)

Estimated Time: 60 minutes

Exam Details

Questions: 28 total (25 multiple choice, 3 short answer)
Time: 60 minutes (recommended)
Topics Covered: Unit 9: Parametric Equations, Polar Coordinates, and Vector-Valued Functions (BC ONLY)
Format: Multiple Choice, Short Answer questions matching real exam difficulty
Scoring: Instant results with detailed explanations

A. Multiple-Choice Questions (25 Questions)

Select the one best answer for each question.

1. [Skill: 1.E | Topic: 9.1] A curve is defined by the parametric equations $x(t) = e^{2t}$ and $y(t) = \sin(3t)$ for all real numbers $t$. What is the slope of the line tangent to the curve at the point where $t = 0$?

(A)0(B)$\frac{2}{3}$(C)$\frac{3}{2}$(D)3

2. [Skill: 1.E | Topic: 9.1] A curve is defined parametrically by $x(t) = t^2$ and $y(t) = t^3 - 3t$. What is the value of $\frac{d^2y}{dx^2}$ at $t = 1$?

(A)$\frac{3}{2}$(B)3(C)6(D)12

3. [Skill: 2.D | Topic: 9.1] For which of the following points $(x, y)$ does the curve defined by $x(t) = t^3 - 3t^2$ and $y(t) = t^2 - 6t$ have a horizontal tangent line?

(A)(-4, -8)(B)(0, 0)(C)(0, -9)(D)(3, -9)

4. A curve is defined by the parametric equations $x = 2t^3$ and $y = 3t^2$ for $t > 0$. What is the value of $\frac{d^2y}{dx^2}$ at the point where $t = 1$?

(A)$-\frac{1}{6}$(B)$-\frac{1}{2}$(C)$\frac{1}{2}$(D)$1$

5. Let $x = f(t)$ and $y = g(t)$ be twice-differentiable functions where $f'(t) \neq 0$. Which of the following expressions represents $\frac{d^2y}{dx^2}$ in terms of $t$?

(A)$\frac{g''(t)}{f''(t)}$(B)$\frac{f'(t)g''(t) - g'(t)f''(t)}{\left(f'(t)\right)^2}$(C)$\frac{f'(t)g''(t) - g'(t)f''(t)}{\left(f'(t)\right)^3}$(D)$\frac{g'(t)f''(t) - f'(t)g''(t)}{\left(f'(t)\right)^3}$

6. A curve is defined parametrically by $x(t) = 2\cos(t)$ and $y(t) = 3\sin(t)$ for $0 < t < 2\pi$. On which of the following intervals is the curve concave up?

(A)$(0, \pi)$(B)$(\pi, 2\pi)$(C)$(0, \frac{\pi}{2}) \cup (\frac{3\pi}{2}, 2\pi)$(D)$(\frac{\pi}{2}, \frac{3\pi}{2})$

7. A curve in the $xy$-plane is defined by the parametric equations $x(t) = t^3 - 2t$ and $y(t) = 4t^2$. Which of the following definite integrals gives the length of the curve over the interval $0 \le t \le 2$?

(A)$\displaystyle \int_0^2 \sqrt{(3t^2 - 2)^2 + (8t)^2} \, dt$(B)$\displaystyle \int_0^2 \sqrt{(t^3 - 2t)^2 + (4t^2)^2} \, dt$(C)$\displaystyle \int_0^2 \left( (3t^2 - 2)^2 + (8t)^2 \right) \, dt$(D)$\displaystyle \int_0^2 \sqrt{3t^2 - 2 + 8t} \, dt$

8. Which of the following integrals gives the length of the curve defined by the parametric equations $x(t) = \sin t - t \cos t$ and $y(t) = \cos t + t \sin t$ for $0 \le t \le \pi$?

(A)$\displaystyle \int_0^\pi \sqrt{1 + t^2} \, dt$(B)$\displaystyle \int_0^\pi t \, dt$(C)$\displaystyle \int_0^\pi t^2 \, dt$(D)$\displaystyle \int_0^\pi \sqrt{2t^2} \, dt$

9. The position of a particle moving in the $xy$-plane is given by the parametric equations $x(t) = e^t \cos t$ and $y(t) = e^t \sin t$. What is the total distance traveled by the particle from $t=0$ to $t=\ln 2$?

(A)$\sqrt{2}$(B)$2\sqrt{2}$(C)$2$(D)$\sqrt{2}(\ln 2)$

10. [Skill: 1.E | Topic: 9.4] The position of a particle moving in the $xy$-plane is given by the vector-valued function $\vec{r}(t) = \langle t e^{-t}, \ln(t^2+1) \rangle$ for $t \ge 0$. What is the velocity vector of the particle at time $t=1$?

(A)$\langle 0, 1 \rangle$(B)$\langle e^{-1}, 1 \rangle$(C)$\langle -e^{-1}, \frac{1}{2} \rangle$(D)$\langle 0, \frac{1}{2} \rangle$

11. [Skill: 1.E | Topic: 9.4] A particle moves in the $xy$-plane such that its position vector at time $t$ is given by $\vec{r}(t) = \langle \cos(2t), 3\sin(t) \rangle$. What is the acceleration vector of the particle at time $t = \frac{pi}{6}$?

(A)$\langle -1, -\frac{3}{2} \rangle$(B)$\langle -2, -\frac{3}{2} \rangle$(C)$\langle -2, \frac{3\sqrt{3}}{2} \rangle$(D)$\langle -4, -\frac{3}{2} \rangle$

12. [Skill: 1.C | Topic: 9.4] The position of an object is defined by the vector-valued function $\vec{r}(t) = \langle 4t, \int_2^{t^2} \sqrt{u+3} \, du \rangle$. Which of the following is the derivative vector $\vec{r}'(1)$?

(A)$\langle 4, 2 \rangle$(B)$\langle 4, 4 \rangle$(C)$\langle 4, 8 \rangle$(D)$\langle 0, 4 \rangle$

13. A particle moves in the $xy$-plane with velocity vector given by $\vec{v}(t) = \langle e^{t/2}, \sin(2t) \rangle$ for time $t \ge 0$. At time $t=0$, the particle is at position $(4, 1)$. Which of the following describes the position vector $\vec{r}(t)$ of the particle?

(A)$\langle $2e^{t/2} + 2, $\; $-$\frac{1}{2}\cos$(2t) + $\frac{3}{2} \rangle$(B)$\langle $2e^{t/2} + 2, $\; $-$\frac{1}{2}\cos$(2t) + 1 $\rangle$(C)$\langle \frac{1}{2}$e^{t/2} + $\frac{7}{2}$, $\; $2$\cos$(2t) - 1 $\rangle$(D)$\langle $e^{t/2} + 3, $\; $-$\cos$(2t) + 2 $\rangle$

14. A particle moves along a curve in the $xy$-plane such that its velocity at time $t > -2$ is given by the vector $\vec{v}(t) = \langle \frac{1}{t+2}, 6t^2 \rangle$. At time $t=0$, the particle is at the point $(3, -1)$. What is the position of the particle at time $t=1$?

(A)$(3 + \ln(3), 1)$(B)$(3 + \ln(1.5), 1)$(C)$(3 + \ln(1.5), 2)$(D)$(\ln(1.5), 2)$

15. The velocity vector of a particle moving in the $xy$-plane is given by $\vec{v}(t) = \langle 2t - 1, 3\sqrt{t} \rangle$ for $t \ge 0$. At time $t=1$, the position of the particle is $(2, 5)$. What is the position vector of the particle at time $t=4$?

(A)$\langle $12, 14 $\rangle$(B)$\langle $14, 19 $\rangle$(C)$\langle $16, 21 $\rangle$(D)$\langle $14, 21 $\rangle$

16. A particle moves along a curve in the $xy$-plane such that its position at time $t \ge 0$ is given by the parametric equations $x(t) = t^2$ and $y(t) = \frac{1}{3}t^3 - t$. What is the speed of the particle at time $t=2$?

(A)3(B)4(C)5(D)7

17. The velocity vector of a particle moving in the $xy$-plane is given by $\vec{v}(t) = \langle \sin(t^2), e^{2t} \rangle$ for $0 \le t \le 3$. Which of the following integrals represents the total distance traveled by the particle over the interval $0 \le t \le 3$?

(A)$\displaystyle \int_0^3 (\sin(t^2) + e^{2t}) \, dt$(B)$\displaystyle \int_0^3 \sqrt{\sin^2(t^2) + e^{4t}} \, dt$(C)$\displaystyle \sqrt{\int_0^3 (\sin^2(t^2) + e^{4t}) \, dt}$(D)$\displaystyle \int_0^3 \sqrt{\sin(t^2) + e^{2t}} \, dt$

18. At time $t=0$, a particle is at position $(2, -1)$. The velocity of the particle at time $t \ge 0$ is given by the vector $\vec{v}(t) = \langle \frac{1}{t+1}, 2t \rangle$. What is the position of the particle at time $t=3$?

(A)$(\ln 4, 9)$(B)$(2 + \ln 4, 8)$(C)$(2 + \ln 4, 9)$(D)$(\ln 4, 8)$

19. Consider the polar curve defined by $r = 2\sin(3\theta)$ for $0 \le \theta \le \pi$. What is the slope of the line tangent to the curve at the point where $\theta = \frac{\pi}{4}$?

(A)-2(B)-1/2(C)1/2(D)2

20. For the polar curve given by $r = 1 + \cos\theta$, which of the following values of $\theta$ in the interval $0 < \theta < \pi$ corresponds to a point where the curve has a vertical tangent line?

(A)$\pi$/3(B)$\pi$/2(C)2$\pi$/3(D)5$\pi$/6

21. The polar curve $r = 2\sin\theta$ represents a circle. What is the instantaneous rate of change of the x-coordinate with respect to $\theta$ at the point where $\theta = \frac{\pi}{3}$?

(A)-1(B)0(C)1(D)$\sqrt{3}$

22. Which of the following integrals gives the area of the region enclosed by one petal of the rose curve defined by the polar function $r = 4\sin(3\theta)$?

(A)$\frac{1}{2} \int_{0}^{\pi/3} (4\sin(3\theta))^2 d\theta$(B)$\int_{0}^{\pi/3} (4\sin(3\theta))^2 d\theta$(C)$\frac{1}{2} \int_{0}^{\pi} (4\sin(3\theta))^2 d\theta$(D)$\frac{1}{2} \int_{0}^{\pi/6} (4\sin(3\theta))^2 d\theta$

23. Consider the limaçon defined by the polar curve $r = 1 + 2\cos\theta$. Which of the following expressions represents the area of the inner loop of the curve?

(A)$\frac{1}{2} \int_{0}^{2\pi} (1 + 2\cos\theta)^2 d\theta$(B)$\frac{1}{2} \int_{2\pi/3}^{4\pi/3} (1 + 2\cos\theta)^2 d\theta$(C)$\int_{2\pi/3}^{4\pi/3} (1 + 2\cos\theta)^2 d\theta$(D)$\frac{1}{2} \int_{0}^{2\pi/3} (1 + 2\cos\theta)^2 d\theta$

24. Which of the following integrals represents the area of the region enclosed inside the circle $r = 6\sin\theta$ and outside the limaçon $r = 2 + 2\sin\theta$?

(A)$\frac{1}{2} \int_{\frac{\pi}{6}}^{\frac{5\pi}{6}} \left( 6\sin\theta - (2 + 2\sin\theta) \right)^2 d\theta$(B)$\frac{1}{2} \int_{\frac{\pi}{6}}^{\frac{5\pi}{6}} \left( (6\sin\theta)^2 - (2 + 2\sin\theta)^2 \right) d\theta$(C)$\int_{\frac{\pi}{6}}^{\frac{5\pi}{6}} \left( (6\sin\theta)^2 - (2 + 2\sin\theta)^2 \right) d\theta$(D)$\frac{1}{2} \int_{0}^{\pi} \left( (6\sin\theta)^2 - (2 + 2\sin\theta)^2 \right) d\theta$

25. Let $R$ be the region inside the polar curve $r = 4\cos\theta$ and inside the polar curve $r = 2$. Which of the following expressions gives the area of $R$?

(A)$\int_{0}^{\frac{\pi}{3}} 2^2 d\theta + \int_{\frac{\pi}{3}}^{\frac{\pi}{2}} (4\cos\theta)^2 d\theta$(B)$\frac{1}{2} \int_{0}^{\frac{\pi}{3}} (2)^2 d\theta + \frac{1}{2} \int_{\frac{\pi}{3}}^{\frac{\pi}{2}} (4\cos\theta)^2 d\theta$(C)$\int_{-\frac{\pi}{3}}^{\frac{\pi}{3}} \frac{1}{2}(2)^2 d\theta + \int_{\frac{\pi}{3}}^{\frac{\pi}{2}} \frac{1}{2}(4\cos\theta)^2 d\theta$(D)$2 \left[ \frac{1}{2} \int_{0}^{\frac{\pi}{3}} (2)^2 d\theta + \frac{1}{2} \int_{\frac{\pi}{3}}^{\frac{\pi}{2}} (4\cos\theta)^2 d\theta \right]$

B. Short Answer Questions (3 Questions)

Answer all parts of each question. Answers must be in essay form. Outlines or lists alone are not acceptable.

Question 26:

Question 27:

Question 28:

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