AP Calculus BC Unit 9: Parametric Equations, Polar Coordinates and Vector-Valued Functions

Curves that are not graphs of one function of $x$: parametric curves, where $x$ and $y$ both depend on $t$; vector-valued functions, which describe motion in the plane; and polar curves, where the distance $r$ from the pole depends on the angle $\theta$. The main skills are slopes, concavity, arc length, motion (velocity, speed, distance) and polar area.

BC topics on this page: 9.1 Parametric curves and dy/dx; 9.2 Second derivative of a parametric curve; 9.3 Arc length of a parametric curve; 9.4 Vector-valued functions and their derivatives; 9.5 Integrating vector-valued functions; 9.6 Motion in the plane; 9.7 Polar coordinates and polar derivatives; 9.8 Area inside one polar curve; 9.9 Area between two polar curves.

Key formulas and facts

• Parametric slope: $\frac{dy}{dx}=\frac{dy/dt}{dx/dt}$ (where $\frac{dx}{dt}\ne0$); horizontal tangent: $\frac{dy}{dt}=0$, $\frac{dx}{dt}\ne0$; vertical tangent: $\frac{dx}{dt}=0$, $\frac{dy}{dt}\ne0$
• Second derivative: $\frac{d^2y}{dx^2}=\dfrac{\frac{d}{dt}\left(\frac{dy}{dx}\right)}{dx/dt}$
• Arc length and distance traveled: $\int_a^b\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}\,dt$
• Motion: $\mathbf v(t)=\mathbf r'(t)$, $\mathbf a(t)=\mathbf v'(t)$, speed $=|\mathbf v(t)|=\sqrt{x'(t)^2+y'(t)^2}$, and $\mathbf r(b)=\mathbf r(a)+\int_a^b\mathbf v(t)\,dt$
• Polar to rectangular: $x=r\cos\theta$, $y=r\sin\theta$, $r^2=x^2+y^2$; polar slope: $\frac{dy}{dx}=\dfrac{r'\sin\theta+r\cos\theta}{r'\cos\theta-r\sin\theta}$
• Polar area: $\frac12\int_\alpha^\beta r^2\,d\theta$; between curves: $\frac12\int_\alpha^\beta\left(r_{\text{outer}}^2-r_{\text{inner}}^2\right)d\theta$, with $\alpha,\beta$ from the intersections
• $\frac{dr}{d\theta}<0$ with $r>0$: the point is moving toward the pole; $\frac{dr}{d\theta}>0$ with $r>0$: moving away

Worked examples

• Slope: $x=t^2+t$, $y=t^3$ at $t=1$: $\frac{dy}{dx}=\frac{3t^2}{2t+1}=1$
• Concavity: $x=t^2$, $y=t^3-t$: $\frac{dy}{dx}=\frac{3t^2-1}{2t}=\frac32t-\frac1{2t}$, so $\frac{d^2y}{dx^2}=\frac{\frac32+\frac1{2t^2}}{2t}$, which is $1$ at $t=1$
• Speed: $\mathbf r(t)=\langle t^2,\ 4t\rangle$ at $t=3$: $\mathbf v=\langle6,4\rangle$, speed $\sqrt{52}=2\sqrt{13}$
• Position: $\mathbf v(t)=\langle2t,\ 1\rangle$, $\mathbf r(0)=\langle3,1\rangle$: $\mathbf r(2)=\langle3+4,\ 1+2\rangle=\langle7,3\rangle$
• Polar area: the circle $r=4\sin\theta$ has area $\frac12\int_0^{\pi}16\sin^2\theta\,d\theta=4\pi$
• Between curves: inside $r=2$ and outside $r=2-2\cos\theta$: the curves meet at $\theta=\pm\frac\pi2$, and the area is $\frac12\int_{-\pi/2}^{\pi/2}\left(4-(2-2\cos\theta)^2\right)d\theta=8-\pi$

Common traps

• Writing $\frac{d^2y}{dx^2}$ as $\frac{d}{dt}\left(\frac{dy}{dx}\right)$ without dividing by $\frac{dx}{dt}$.
• Using $\frac{dy}{dt}=0$ alone for a horizontal tangent without checking that $\frac{dx}{dt}\ne0$ there.
• Confusing displacement $\int\mathbf v\,dt$ with distance traveled $\int|\mathbf v|\,dt$.
• Using $\frac{dr}{d\theta}$ as the slope of a polar curve: the slope is $\frac{dy}{dx}$, from $x=r\cos\theta$ and $y=r\sin\theta$.
• Polar area of the region between curves as $\frac12\int(r_1-r_2)^2\,d\theta$: square each radius first, then subtract.
• Integrating over a full turn when the curve traces part of the region twice (roses), or missing an inner loop where $r<0$.
• Giving $\theta$ in the wrong quadrant from $\tan\theta=\frac yx$: check the signs of $x$ and $y$.

Practice questions

63 multiple-choice questions and 3 free-response questions. Work each one out before you open the answer. In a free-response scoring guide, each line is worth the points it states. A free account opens the worked answers and scoring guides of 20 questions a day; Pro opens every one.

Question 1 · Multiple choice

The parametric equations $x=t^2+3$, $y=t^3-3t$ trace a curve in the plane. Find $\dfrac{dy}{dx}$ when $t=2$.

1. $\frac{4}{9}$
2. $9$
3. $\frac{9}{4}$
4. $\frac{3}{2}$

Question 2 · Multiple choice

Which of the following is an equation of the line tangent to the curve $x=t^3$, $y=t^2+2t$ at the point where $t=1$?

1. $y=\frac34x+\frac94$
2. $y=\frac43x+\frac53$
3. $y=4x-1$
4. $y=\frac43x+3$

Question 3 · Multiple choice

At which points does the curve $x=t^2-6t$, $y=t^3-12t$ have a horizontal tangent line?

1. $(-8,16)$ and $(16,-16)$
2. $(-9,-9)$ only
3. $(-8,-16)$ only
4. $(-8,-16)$ and $(16,16)$

Question 4 · Multiple choice

The curve $x=t^3-3t$, $y=t^2$ has vertical tangent lines at which values of $x$?

1. $x=0$ only
2. $x=-1$ and $x=1$
3. $x=-2$ and $x=2$
4. $x=1$ only

Question 5 · Multiple choice

For the curve $x=\ln t$, $y=t^2+1$ with $t>0$, which of the following is $\dfrac{dy}{dx}$ in terms of $t$?

1. $2t^2$
2. $2t$
3. $\frac{2}{t}$
4. $\frac{1}{2t^2}$

Question 6 · Multiple choice

A curve is given by $x=\sin t$, $y=\cos(2t)$. What is the slope of the curve at $t=\dfrac{\pi}{6}$?

1. $2$
2. $- \sqrt{3}$
3. $- \frac{1}{2}$
4. $-2$

Question 7 · Multiple choice

A particle moves along a curve with $x=x(t)$ and $y=y(t)$. The table gives values of $x'(t)$ and $y'(t)$. What is the slope of the curve at the point where $t=3$?

1. $- \frac{2}{3}$
2. $- \frac{3}{2}$
3. $\frac{5}{2}$
4. $\frac{3}{2}$

Question 8 · Multiple choice

For the curve $x=t^2$, $y=t^3$, what is $\dfrac{d^2y}{dx^2}$ at $t=1$?

1. $\frac{3}{2}$
2. $3$
3. $6$
4. $\frac{3}{4}$

Question 9 · Multiple choice

The curve $x=2t$, $y=t^3-t^2$ is concave up for which values of $t$?

1. $t<\frac13$
2. $t>\frac13$
3. $t>\frac23$
4. $0<t<\frac23$

Question 10 · Multiple choice

For the curve $x=e^t$, $y=te^t$, which of the following is $\dfrac{d^2y}{dx^2}$?

1. $e^{-t}$
2. $1$
3. $e^{t}$
4. $(1+t)e^{-t}$

Question 11 · Multiple choice

A particle moves on the circle $x=\cos t$, $y=\sin t$. What is $\dfrac{d^2y}{dx^2}$ at $t=\dfrac{\pi}{4}$?

1. $2 \sqrt{2}$
2. $-1$
3. $- 2 \sqrt{2}$
4. $- \sqrt{2}$

Question 12 · Multiple choice

For the curve $x=t^3+t$, $y=t^2$, what is $\dfrac{d^2y}{dx^2}$ at $t=0$?

1. $0$
2. $2$
3. $1$
4. $-2$

Question 13 · Multiple choice

For the curve $x=t^2$, $y=t^4+t$ with $t>0$, what is $\dfrac{d^2y}{dx^2}$ at $t=1$?

1. $\frac{7}{2}$
2. $\frac{5}{2}$
3. $6$
4. $\frac{7}{4}$

Question 14 · Multiple choice

For the curve $x=3t$, $y=\ln t$ with $t>0$, what is $\dfrac{d^2y}{dx^2}$ at $t=1$?

1. $- \frac{1}{3}$
2. $\frac{1}{3}$
3. $- \frac{1}{9}$
4. $-1$

Question 15 · Multiple choice

What is the length of the curve $x=t^2$, $y=\frac23t^3$ for $0\le t\le2\sqrt2$?

1. $\frac{52}{3}$
2. $18$
3. $\frac{26}{3}$
4. $16$

Question 16 · Multiple choice

What is the length of the path $x=3\cos t$, $y=3\sin t$ for $0\le t\le\pi$?

1. $3 \pi$
2. $6 \pi$
3. $\frac{9 \pi}{2}$
4. $6$

Question 17 · Multiple choice

Which integral gives the length of the curve $x=e^t$, $y=t^2$ for $0\le t\le1$?

1. $\displaystyle\int_0^1\sqrt{e^{t}+2t}\,dt$
2. $\displaystyle\int_0^1\sqrt{1+4t^2e^{-2t}}\,dt$
3. $\displaystyle\int_0^1\sqrt{e^{2t}+4t^2}\,dt$
4. $\displaystyle\int_0^1\left(e^{2t}+4t^2\right)dt$

Question 18 · Multiple choice

What is the length of one arch of the cycloid $x=t-\sin t$, $y=1-\cos t$, for $0\le t\le2\pi$?

1. $2 \pi$
2. $8$
3. $4$
4. $4 \pi$

Question 19 · Multiple choice

What is the length of the curve $x=t^3-3t$, $y=t^2$ for $0\le t\le2$?

1. $5.657$
2. $6.0$
3. $8.944$
4. $7.605$

Question 20 · Multiple choice

What is the length of the part of the curve $x=\cos^3t$, $y=\sin^3t$ with $0\le t\le\dfrac{\pi}{2}$?

1. $\frac{3}{4}$
2. $\frac{\pi}{2}$
3. $\frac{3}{2}$
4. $\sqrt{2}$

Question 21 · Multiple choice

A curve is given by $x=\ln t$, $y=t$ for $1\le t\le3$. What is its length?

1. $2.302$
2. $2.0$
3. $2.299$
4. $3.099$

Question 22 · Multiple choice

The position of a particle is $\mathbf r(t)=\langle t^3-2t,\ e^{2t}\rangle$. What is its velocity vector at $t=0$?

1. $\langle -2,\,1\rangle$
2. $\langle 0,\,1\rangle$
3. $\langle 1,\,2\rangle$
4. $\langle -2,\,2\rangle$

Question 23 · Multiple choice

A particle has position $\mathbf r(t)=\langle t^4,\ \ln t\rangle$ for $t>0$. What is its acceleration vector at $t=1$?

1. $\langle 4,\,1\rangle$
2. $\langle 12,\,-1\rangle$
3. $\langle 12,\,1\rangle$
4. $\langle 24,\,2\rangle$

Question 24 · Multiple choice

A particle moves with position $\mathbf r(t)=\langle3t,\ t^2-1\rangle$. What is its speed at $t=2$?

1. $7$
2. $\sqrt{13}$
3. $25$
4. $5$

Question 25 · Multiple choice

For $\mathbf r(t)=\langle\cos(2t),\ \sin(3t)\rangle$, what is $\mathbf r'\left(\frac{\pi}{4}\right)$?

1. $\langle -2,\,\frac{3\sqrt2}{2}\rangle$
2. $\langle -2,\,-\frac{3\sqrt2}{2}\rangle$
3. $\langle -1,\,-\frac{\sqrt2}{2}\rangle$
4. $\langle 2,\,-\frac{3\sqrt2}{2}\rangle$

Question 26 · Multiple choice

A particle has position $\mathbf r(t)=\langle e^t,\ t^2-4t\rangle$. At what time $t$ is its velocity vector horizontal?

1. $2$
2. $0$
3. $4$
4. $\ln{\left(2 \right)}$

Question 27 · Multiple choice

A particle has position $\mathbf r(t)=\langle\sqrt t,\ t^2+1\rangle$ for $t>0$. What is the slope of its path at $t=1$?

1. $1$
2. $\frac{1}{4}$
3. $4$
4. $2$

Question 28 · Multiple choice

A particle moves with position $\mathbf r(t)=\langle4\cos t,\ 4\sin t\rangle$. What is the magnitude of its acceleration at any time $t$?

1. $16$
2. $4$
3. $1$
4. $0$

Question 29 · Multiple choice

A particle has velocity $\mathbf v(t)=\langle2t,\ 3t^2\rangle$ and position $\mathbf r(0)=\langle1,-2\rangle$. What is $\mathbf r(2)$?

1. $\langle 4,\,8\rangle$
2. $\langle 5,\,10\rangle$
3. $\langle 3,\,6\rangle$
4. $\langle 5,\,6\rangle$

Question 30 · Multiple choice

What is $\displaystyle\int_0^1\langle e^t,\ 2t\rangle\,dt$?

1. $\langle e,\,1\rangle$
2. $\langle e-1,\,2\rangle$
3. $\langle e-1,\,1\rangle$
4. $\langle e+1,\,1\rangle$

Question 31 · Multiple choice

A particle has acceleration $\mathbf a(t)=\langle6t,\ 2\rangle$, initial velocity $\mathbf v(0)=\langle0,1\rangle$ and initial position $\mathbf r(0)=\langle0,0\rangle$. What is $\mathbf r(1)$?

1. $\langle 1,\,2\rangle$
2. $\langle 3,\,2\rangle$
3. $\langle 1,\,1\rangle$
4. $\langle 3,\,3\rangle$

Question 32 · Multiple choice

A particle starts at the origin with velocity $\mathbf v(t)=\langle\cos t,\ \sin t\rangle$. Where is it at $t=\pi$?

1. $\langle 0,\,2\rangle$
2. $\langle 0,\,0\rangle$
3. $\langle -1,\,0\rangle$
4. $\langle 2,\,0\rangle$

Question 33 · Multiple choice

What is $\displaystyle\int_1^4\left\langle\frac1t,\ \sqrt t\right\rangle dt$?

1. $\langle \ln4,\,\frac{16}{3}\rangle$
2. $\langle \frac34,\,\frac{14}{3}\rangle$
3. $\langle \ln4,\,\frac{14}{3}\rangle$
4. $\langle \ln3,\,\frac{14}{3}\rangle$

Question 34 · Multiple choice

A particle has velocity $\mathbf v(t)=\langle t-1,\ 2\rangle$. What is its displacement vector from $t=0$ to $t=3$?

1. $\langle \frac52,\,6\rangle$
2. $\langle \frac32,\,6\rangle$
3. $\langle \frac32,\,2\rangle$
4. $\langle 2,\,6\rangle$

Question 35 · Multiple choice

A particle moves with velocity $\mathbf v(t)=\langle\ln(1+t^2),\ e^{-t}\rangle$, and $x(0)=2$. What is $x(3)$?

1. $3.406$
2. $2.303$
3. $4.303$
4. $5.406$

Question 36 · Multiple choice

A particle moves with $x(t)=t^3$ and $y(t)=2t^2$. What is its speed at $t=2$?

1. $20$
2. $4 \sqrt{5}$
3. $4 \sqrt{13}$
4. $8 \sqrt{2}$

Question 37 · Multiple choice

A particle moves with $\dfrac{dx}{dt}=\cos(t^2)$ and $\dfrac{dy}{dt}=e^{t/2}$. What is the total distance the particle travels from $t=0$ to $t=2$?

1. $3.828$
2. $3.437$
3. $2.0$
4. $4.207$

Question 38 · Multiple choice

A particle moves in the plane with $x'(t)=t^2-4t+3$. For which times $t$ is the particle moving to the left?

1. $t<1$ and $t>3$
2. $t>3$
3. $0<t<2$
4. $1<t<3$

Question 39 · Multiple choice

A particle moves with $x'(t)=t^2-1$ and $y'(t)=t^2-3t+2$ for $t\ge0$. At what time is the particle at rest?

1. $2$
2. $1$
3. $0$
4. $\frac{3}{2}$

Question 40 · Multiple choice

A particle moves with velocity $\mathbf v(t)=\langle2t-3,\ \sqrt{t+1}\rangle$. What is the magnitude of its displacement from $t=0$ to $t=3$?

1. $\frac{\sqrt{205}}{3}$
2. $\frac{9}{2}$
3. $\frac{16}{3}$
4. $\frac{14}{3}$

Question 41 · Multiple choice

A particle moves with velocity $\mathbf v(t)=\langle e^{\sin t},\ t\cos t\rangle$. What is its speed at $t=1.5$?

1. $2.711$
2. $2.714$
3. $2.818$
4. $0.106$

Question 42 · Multiple choice

A particle has velocity $\mathbf v(t)=\langle3,\ 4t\rangle$ and is at $(1,0)$ when $t=0$. How far is the particle from the origin at $t=2$?

1. $\sqrt{113}$
2. $10$
3. $5$
4. $\sqrt{89}$

Question 43 · Multiple choice

The point with polar coordinates $(r,\theta)=\left(4,\dfrac{2\pi}{3}\right)$ has which rectangular coordinates?

1. $(2,\,-2\sqrt3)$
2. $(-2\sqrt3,\,2)$
3. $(-2,\,2\sqrt3)$
4. $(2,\,2\sqrt3)$

Question 44 · Multiple choice

The point $(-3,3)$ is written in polar form $(r,\theta)$ with $r>0$ and $0\le\theta<2\pi$. Which is it?

1. $\left(3\sqrt2,\,-\frac{\pi}{4}\right)$
2. $\left(3\sqrt2,\,\frac{3\pi}{4}\right)$
3. $\left(3\sqrt2,\,\frac{7\pi}{4}\right)$
4. $\left(18,\,\frac{3\pi}{4}\right)$

Question 45 · Multiple choice

What is the slope of the tangent line to the polar curve $r=2+2\cos\theta$ at $\theta=\dfrac{\pi}{2}$?

1. $-1$
2. $0$
3. $2$
4. $1$

Question 46 · Multiple choice

What is the slope of the tangent line to the spiral $r=\theta$ at $\theta=\pi$?

1. $- \pi$
2. $\frac{1}{\pi}$
3. $\pi$
4. $0$

Question 47 · Multiple choice

A point moves along the polar curve $r=1+2\sin\theta$ as $\theta$ increases. At $\theta=\dfrac{3\pi}{4}$, which statement is true?

1. The point is moving toward the pole, because $r>0$ and $\frac{dr}{d\theta}<0$.
2. The point is moving away from the pole, because $r>0$ and $\frac{dr}{d\theta}>0$.
3. The point is moving away from the pole, because $r>0$ and $\frac{dr}{d\theta}<0$.
4. The point is at the pole, because $r=0$.

Question 48 · Multiple choice

The polar curve $r=6\cos\theta$ is a circle. Which rectangular equation describes it?

1. $(x-3)^2+y^2=9$
2. $x^2+(y-3)^2=9$
3. $(x-6)^2+y^2=36$
4. $(x-3)^2+y^2=36$

Question 49 · Multiple choice

What is $\dfrac{dy}{dx}$ for the polar curve $r=2+\sin\theta$ at $\theta=0$?

1. $\frac{1}{2}$
2. $1$
3. $2$
4. $-2$

Question 50 · Multiple choice

The figure shows the cardioid $r=2+2\cos\theta$. What is the area of the region it encloses?

1. $12 \pi$
2. $6 \pi$
3. $4 \pi$
4. $3 \pi$

Question 51 · Multiple choice

What is the area of one petal of the rose $r=4\sin(3\theta)$?

1. $\frac{8 \pi}{3}$
2. $4 \pi$
3. $\frac{2 \pi}{3}$
4. $\frac{4 \pi}{3}$

Question 52 · Multiple choice

What is the area of the region bounded by the spiral $r=\theta$ for $0\le\theta\le\pi$ and the $x$-axis?

1. $\frac{\pi^{3}}{3}$
2. $\frac{\pi^{2}}{2}$
3. $\frac{\pi^{3}}{6}$
4. $\frac{\pi^{3}}{2}$

Question 53 · Multiple choice

Which integral gives the area of the part of the region inside $r=1+\cos\theta$ that lies in the first quadrant?

1. $\displaystyle\frac12\int_0^{\pi/2}(1+\cos\theta)^2\,d\theta$
2. $\displaystyle\int_0^{\pi/2}(1+\cos\theta)\,d\theta$
3. $\displaystyle\frac12\int_0^{\pi}(1+\cos\theta)^2\,d\theta$
4. $\displaystyle\int_0^{\pi/2}(1+\cos\theta)^2\,d\theta$

Question 54 · Multiple choice

The figure shows the limaçon $r=1+2\cos\theta$. What is the area of the region inside its small inner loop?

1. $\frac{3 \sqrt{3}}{2} + 2 \pi$
2. $\frac{3 \sqrt{3}}{2} + \pi$
3. $\frac{\pi}{2}$
4. $\pi - \frac{3 \sqrt{3}}{2}$

Question 55 · Multiple choice

What is the area of the region swept out by the curve $r=e^{\theta/4}$ for $0\le\theta\le2\pi$ (bounded by the curve and the segment of the positive $x$-axis joining its ends)?

1. $-2 + 2 e^{\pi}$
2. $-1 + e^{\pi}$
3. $- \frac{1}{2} + \frac{e^{\pi}}{2}$
4. $-4 + 4 e^{\frac{\pi}{2}}$

Question 56 · Multiple choice

What is the area of the region enclosed by the polar curve $r=2+\cos^3\theta$?

1. $12.566$
2. $14.137$
3. $6.774$
4. $13.548$

Question 57 · Multiple choice

What is the area of the region inside the circle $r=3\cos\theta$ and outside the cardioid $r=1+\cos\theta$?

1. $\frac{\pi}{2}$
2. $\pi$
3. $2 \pi$
4. $\frac{3 \pi}{2}$

Question 58 · Multiple choice

The circles $r=2\sin\theta$ and $r=2\cos\theta$ meet at the pole and at one other point. What is $\theta$ at the other point, for $0<\theta<\dfrac{\pi}{2}$?

1. $\frac{\pi}{4}$
2. $\frac{\pi}{3}$
3. $\frac{\pi}{6}$
4. $\frac{\pi}{2}$

Question 59 · Multiple choice

What is the area of the region inside both circles $r=2\sin\theta$ and $r=2\cos\theta$?

1. $\frac{\pi}{4}$
2. $1 + \frac{\pi}{2}$
3. $-1 + \frac{\pi}{2}$
4. $-2 + \pi$

Question 60 · Multiple choice

What is the area of the region inside $r=4\sin\theta$ and outside $r=2$?

1. $- 2 \sqrt{3} + \frac{4 \pi}{3}$
2. $2 \sqrt{3} + \frac{4 \pi}{3}$
3. $\frac{2 \pi}{3} + 2 \sqrt{3}$
4. $\frac{8 \pi}{3}$

Question 61 · Multiple choice

Which integral gives the area of the region inside $r=1+\cos\theta$ and outside $r=1$?

1. $\displaystyle\frac12\int_{-\pi/2}^{\pi/2}\cos^2\theta\,d\theta$
2. $\displaystyle\frac12\int_{0}^{2\pi}\left((1+\cos\theta)^2-1\right)d\theta$
3. $\displaystyle\int_{-\pi/2}^{\pi/2}\cos\theta\,d\theta$
4. $\displaystyle\frac12\int_{-\pi/2}^{\pi/2}\left((1+\cos\theta)^2-1\right)d\theta$

Question 62 · Multiple choice

The curves $r=2+\cos\theta$ and $r=4\sin\theta$ intersect at one point in the first quadrant. To the nearest thousandth, what is $\theta$ at that point?

1. $0.524$
2. $1.047$
3. $0.751$
4. $0.785$

Question 63 · Multiple choice

What is the area of the region inside the cardioid $r=2+2\cos\theta$ and outside the circle $r=2$?

1. $\pi + 8$
2. $8 - \pi$
3. $\pi + 4$
4. $2 \pi + 8$

Question 64 · Free response (9 points)

A particle moves in the $xy$-plane for $0\le t\le4$ with velocity $\mathbf v(t)=\langle t\sin t,\ e^{-t}+1\rangle$. At $t=0$ the particle is at $(2,-1)$.

1. (a) Find the speed of the particle at $t=2$. [1 point]
2. (b) Find the acceleration vector of the particle at $t=2$. [2 points]
3. (c) Find the total distance the particle travels from $t=0$ to $t=4$. [2 points]
4. (d) Find the position of the particle at $t=4$. [2 points]
5. (e) For what values of $t$ in $0\le t\le4$ is the particle moving to the left? Give a reason. [2 points]

Question 65 · Free response (9 points)

A curve $C$ is defined by $x=t^2$ and $y=\frac13t^3-t$ for $t\ge0$.

1. (a) Find $\dfrac{dy}{dx}$ in terms of $t$, and its value at $t=2$. [2 points]
2. (b) Find the point on $C$ where the tangent line is horizontal. [2 points]
3. (c) Find $\dfrac{d^2y}{dx^2}$ in terms of $t$. Is $C$ concave up or concave down at $t=2$? Justify your answer. [3 points]
4. (d) Find the length of $C$ from $t=0$ to $t=2$. [2 points]

Question 66 · Free response (9 points)

The figure shows the circle $r=3$ and the cardioid $r=2+2\cos\theta$.

1. (a) Find the area of the region inside the cardioid and outside the circle. [2 points]
2. (b) Find $\dfrac{dr}{d\theta}$ for the cardioid at $\theta=\dfrac{\pi}{2}$. What does it tell you about a point moving along the cardioid as $\theta$ increases through $\dfrac{\pi}{2}$? [2 points]
3. (c) Find the slope of the line tangent to the cardioid at $\theta=\dfrac{\pi}{2}$. [2 points]
4. (d) Find the area of the region inside both the circle and the cardioid. [3 points]