Parametric equations, polar coordinates and vector-valued functions

Practice sheet

Fresh questions from the whole chapter, to do on paper.

New questions Include answers

To save a PDF, choose “Save as PDF” in the print dialog.

$x = t^2$ , $y = t^3 + 9t$ . Find $\dfrac{dy}{dx}$ at $t = 3$ .
$x = 3\cos t$ , $y = 3\sin t$ . Find $\dfrac{dy}{dx}$ at $t = \dfrac{\pi}{ 3 }$ . Give an exact value.
$x = t^2$ , $y = t^3 - 27t$ . For which $t > 0$ is the tangent horizontal?
$x = t^2$ , $y = 8t^3$ for $t > 0$ . Find $\dfrac{d^2y}{dx^2}$ in terms of $t$ .
$x = t^2$ , $y = 9t^3$ . Find $\dfrac{d^2y}{dx^2}$ at $t = 3$ .
$x = e^t$ , $y = e^{-3t}$ . Find $\dfrac{d^2y}{dx^2}$ in terms of $t$ .
Find the length of the curve $x = 3\cos t$ , $y = 3\sin t$ , $0 \le t \le 2\pi$ .
Find the length of the path $x = 3t$ , $y = 6t$ , $0 \le t \le 1$ . Give an exact value.
Find the length of $x = t$ , $y = \tfrac{2}{3}t^{3/2}$ for $0 \le t \le 3$ .
A particle moves with position $\vec r(t) = \langle t^2, 6t \rangle$ . Find its speed at $t = 2$ . Give an exact value.
$\vec r(t) = \langle 6t^2, 8\sin t \rangle$ . Find the magnitude of the acceleration at $t = 0$ .
$\vec v(t) = \langle 2t, 3 \rangle$ and $\vec r(0) = \langle 1, 0 \rangle$ . Find the $x$ -coordinate of the position at $t = 2$ .
Find the slope $\dfrac{dy}{dx}$ of the polar curve $r = 8 + \cos\theta$ at $\theta = \dfrac{\pi}{2}$ .
Find the slope of the spiral $r = 4\theta$ at $\theta = \dfrac{\pi}{2}$ .
Where does the polar curve $r = f(\theta)$ , for instance $r = 1 + \sin\theta$ , have a horizontal tangent?
1. Where $\dfrac{dy}{d\theta} = 0$ and $\dfrac{dx}{d\theta} \ne 0$
2. Where $\dfrac{dr}{d\theta} = 0$
3. Where $\dfrac{dx}{d\theta} = 0$
Where does the polar curve $r = f(\theta)$ , for instance $r = 7 + \sin\theta$ , have a horizontal tangent?
1. Where $\dfrac{dy}{d\theta} = 0$ and $\dfrac{dx}{d\theta} \ne 0$
2. Where $\dfrac{dr}{d\theta} = 0$
3. Where $\dfrac{dx}{d\theta} = 0$
Where does the polar curve $r = f(\theta)$ , for instance $r = 6 + \sin\theta$ , have a horizontal tangent?
1. Where $\dfrac{dy}{d\theta} = 0$ and $\dfrac{dx}{d\theta} \ne 0$
2. Where $\dfrac{dr}{d\theta} = 0$
3. Where $\dfrac{dx}{d\theta} = 0$
Where does the polar curve $r = f(\theta)$ , for instance $r = 8 + \sin\theta$ , have a horizontal tangent?
1. Where $\dfrac{dy}{d\theta} = 0$ and $\dfrac{dx}{d\theta} \ne 0$
2. Where $\dfrac{dr}{d\theta} = 0$
3. Where $\dfrac{dx}{d\theta} = 0$
Where does the polar curve $r = f(\theta)$ , for instance $r = 2 + \sin\theta$ , have a horizontal tangent?
1. Where $\dfrac{dy}{d\theta} = 0$ and $\dfrac{dx}{d\theta} \ne 0$
2. Where $\dfrac{dr}{d\theta} = 0$
3. Where $\dfrac{dx}{d\theta} = 0$
Where does the polar curve $r = f(\theta)$ , for instance $r = 5 + \sin\theta$ , have a horizontal tangent?
1. Where $\dfrac{dy}{d\theta} = 0$ and $\dfrac{dx}{d\theta} \ne 0$
2. Where $\dfrac{dr}{d\theta} = 0$
3. Where $\dfrac{dx}{d\theta} = 0$
Where does the polar curve $r = f(\theta)$ , for instance $r = 9 + \sin\theta$ , have a horizontal tangent?
1. Where $\dfrac{dy}{d\theta} = 0$ and $\dfrac{dx}{d\theta} \ne 0$
2. Where $\dfrac{dr}{d\theta} = 0$
3. Where $\dfrac{dx}{d\theta} = 0$
Where does the polar curve $r = f(\theta)$ , for instance $r = 4 + \sin\theta$ , have a horizontal tangent?
1. Where $\dfrac{dy}{d\theta} = 0$ and $\dfrac{dx}{d\theta} \ne 0$
2. Where $\dfrac{dr}{d\theta} = 0$
3. Where $\dfrac{dx}{d\theta} = 0$
Where does the polar curve $r = f(\theta)$ , for instance $r = 3 + \sin\theta$ , have a horizontal tangent?
1. Where $\dfrac{dy}{d\theta} = 0$ and $\dfrac{dx}{d\theta} \ne 0$
2. Where $\dfrac{dr}{d\theta} = 0$
3. Where $\dfrac{dx}{d\theta} = 0$
Use $A = \tfrac{1}{2}\displaystyle\int r^2\,d\theta$ to find the area inside $r = 9$ .
Find the area of one petal of the rose $r = 3\cos(2\theta)$ (the petal for $-\tfrac{\pi}{4} \le \theta \le \tfrac{\pi}{4}$ ).
Find the area inside the cardioid $r = 5(1 + \cos\theta)$ .

Answers

$6$
$- \frac{\sqrt{3}}{3}$
$3$
$\frac{6}{t}$
$\frac{9}{4}$
$12 e^{- 5 t}$
$6 \pi$
$3 \sqrt{5}$
$\frac{14}{3}$
$2 \sqrt{13}$
$12$
$5$
$\frac{1}{8}$
$- \frac{2}{\pi}$
Where $\dfrac{dy}{d\theta} = 0$ and $\dfrac{dx}{d\theta} \ne 0$
$81 \pi$
$\frac{9 \pi}{8}$
$\frac{75 \pi}{2}$