9.4.1How can I plot a polar curve?

Graphs of Polar Equations

9-104.

PILAR’S COORDINATES

Pilar is thinking of a point on the graph at right. All she will tell you is that it is located on a ray rotated $45^\circ$ counterclockwise from the positive $x$ -axis, and $3$ units from the origin.

On polar graph paper, locate Pilar’s point and label it $P$ .
Given a point in polar coordinates $(r,θ)$ , $r$ represents the distance from the pole (origin) and $θ$ represents the counterclockwise angle from the $x$ -axis.

Pilar’s point can be written $( 3 , \frac { \pi } { 4 } )$ . Using this polar notation, locate and label points $R(2,\frac{\pi}{6})$ and $Q(5,\frac{3\pi}{2})$ .
What happens when an angle is negative? Discuss this with your team as you place the points $K(1,-\frac{3\pi}{4})$ and $L(1.5,-\frac{5\pi}{3})$ on your graph.
Pilar’s point can also be written as $( - 3 , - \frac { 3 \pi } { 4 } )$ . Why?

Your teacher will provide you with a model.

9-105.

POLAR WALK

With a rope, create human graphs for the polar curves listed below. While creating the graphs, consider to the following:

$\Rightarrow$ What is the shape of the curve?

$\Rightarrow$ What minimum interval of $θ$ will create a complete graph?

$r = 6$

$r = 9\sin(θ)$

$r = 3 + 7\cos(θ)$

$r = 8\cos(2θ))$

9-106.

What do you think the graph of $r = 5\cos(θ)$ will look like? How will this differ from the graph of $r = -5\cos(θ)$ ? Check your conjecture with your graphing calculator.

9-107.

The region under $y =\sqrt { x - 1 }$ over the interval $1 ≤ x ≤ a$ for some $a > 1$ is rotated about the line $y = −1$ . If the volume of the resulting solid is $10π$ cubic units, what is the value of $a$ ? Homework Help ✎

9-108.

Multiple Choice: If $f(x) = 3x^{1/3} − 4x$ and $x_1 = 1$ , then using Newton’s Method of approximating roots will result in this approximate value for $x_2$ : Homework Help ✎

$0.3333$

$0.5$

$0.6495$

$0.6496$

$0.6667$

9-109.

Examine the integrals below. Consider the multiple tools available for integrating and use the best strategy for each part. Evaluate each integral and briefly describe your method. Homework Help ✎

$\int _ { 0 } ^ { \pi } \frac { \operatorname { sin } ( x ) } { 2 - \operatorname { cos } ( x ) } d x$

$\int _ { - 6 } ^ { 2 } | x | d x$

$\int x ^ { 2 } e ^ { 4 x } d x$

$\int _ { 0 } ^ { \infty } \frac { x } { x ^ { 2 } + 4 } d x$

9-110.

Use the graph of $y = f(x)$ below to sketch $f^\prime(x)$ . Homework Help ✎

9-111.

Calculate the area of the region which includes the origin and is bounded by the curves $f(x) = x + 1$ , $g(x) = x^2 - 1$ , and $h(x) = \cos^{–1}(x)$ . Homework Help ✎

9-112.

No calculator! Evaluate the following limits. Homework Help ✎

$\lim\limits _ { x \rightarrow \infty } \frac { 3 x ^ { 5 } - 2 x ^ { 2 } + 1 } { 2 x ^ { 6 } + 4 x - 1 }$

$\lim\limits _ { x \rightarrow - \infty } e ^ { - 2 x }$

$\lim\limits _ { x \rightarrow 0 } \frac { \operatorname { cot } ( x ) } { \operatorname { ln } ( x ) }$

$\lim\limits _ { n \rightarrow \infty } ( \displaystyle\sum _ { k = 1 } ^ { n } ( \frac { 1 } { 3 } ) ^ { k } )$

Compute without a calculator

9-113.

Consider each of the infinite series below. For each series, decide if it converges or diverges and justify your conclusion. If the series converges, calculate its sum. Homework Help ✎

$\frac { 2 } { 3 } + \frac { 1 } { 3 } + \frac { 1 } { 6 } + \frac { 1 } { 12 } + \ldots$

$0.9 + 0.99 + 0.999 + 0.9999 + …$

$0.1 + 0.12 + 0.123 + 0.1234 + …$

$256 − 128 + 64 − 32 + …$

9-114.

For each of the rectangular coordinates pairs below, write two equivalent polar coordinate pairs. Homework Help ✎

$(3,−3)$

$(0,0)$

$(0,−5)$

$\left(−2,2\sqrt{3}\right)$

9-115.

In the diagram below, $ABCD$ is a parallelogram. Write each of the unlabeled vectors in terms of $\mathbf{p}$ and $\mathbf{q}$ . Here is one to get you started: $\overrightarrow { D C }= \mathbf{q}$ . Homework Help ✎

Parallelogram, A B C D, with the following rays: left side labeled, P, going from A to d, bottom side, labeled, q, going from A to b, diagonal from A to c, & diagonal from d to b.

Calculus Third Edition