Derivatives of Parametric, Polar, and Vector Functions

Practice Questions

AP Calculus BC › Derivatives of Parametric, Polar, and Vector Functions

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1

Solve for $\frac{dy}{dx}$ if and .

2

What is the derivative of $r=7sin\theta-6$ ?

3

Let

$\vec{u}=(\sqrt{1+t^2},2\sqrt{1+t^2},3\sqrt{1+t^2},\cdots n\sqrt{1+t^2})$

What is the derivative of $\vec{u}$ ?

4

Given that $f: \mathbb{R}^{n}}\rightarrow \mathbb{R}$ . We define its gradient as :

$abla f =\left[\frac{\partial f}{\partial x_{1}} \cdots \frac{\partial f}{\partial x_{n}}\right]$

Let $f: \mathbb{R}^{n}}\rightarrow \mathbb{R}$ be given by:

$f(x_{1},x_{2},\cdots, x_{n}})=\sqrt{{x_{1}^2}+x_{2}^2+\cdots+x_{n}^2}}$

What is the gradient of ?

5

Find the derivative $\frac{dy}{dx}$ of the polar function $r(\theta)=3-4sin(\theta)$ .

6

Find the derivative of the following polar equation:

$r=3+8sin\theta$

7

Screen shot 2016 03 30 at 4.58.19 pm

A particle moves around the xy plane such that its position as a function of time is given by the parametric function:

$\begin{pmatrix} x(t)\y(t) \end{pmatrix}=\begin{pmatrix} \sin(3t)\\sin(2t) \end{pmatrix}$ .

What is the slope, $\frac{dy}{dx}$ , of the particle's trajectory when $t=\pi/12$ ?

8

Let $f:\mathbb{R}^{n}\mapsto \mathbb{R}$ .

We define the gradient of as:

$abla f =\left[\frac{\partial f}{\partial x_{1}} \cdots \frac{\partial f}{\partial x_{n}}\right]$

Let $f=sin(x_{1}+2x_{2}+\cdots nx_{n})$ .

Find the vector gradient.

9

Find the derivative of the following set of parametric equations:

10

Find the first derivative of the polar function

$r(\theta)=sin(3\theta)$ .

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