9.4.2What is the polar equation for a rose?

Converting Between Polar and Rectilinear Form

9-116.

For each polar equation below, create a table of values with $\theta = 0 , \frac { \pi } { 12 } , \frac { \pi } { 6 } , \frac { \pi } { 4 } , \frac { \pi } { 3 } , \ldots , 2 \pi$ . Then, on polar graph paper, carefully plot the points for each polar curve on its own set of axes.

CARDIOID: $r=3+3\cos(θ)$

LIMAÇON: $r=2+4\sin(θ)$

ROSE: $r = 5\sin(3θ)$

9-117.

The cardioid and limaçon you graphed in the previous problem are two of the more interesting curves that can be created using polar equations. Below are some polar equations whose graphs take on a more familiar appearance. Use your graphing calculator to graph each of the polar equations below over the given interval. Then, sketch the graph on polar graph paper and identify the familiar shape.

$r =\frac { 4 } { \operatorname { sin } ( \theta ) }$ for $0 ≤ θ < π$

$r =\frac { 2 \operatorname { sin } ( \theta ) + 2 } { \operatorname { cos } ^ { 2 } ( \theta ) }$ for $0 ≤ θ < 2π$

$r = 5\cos(θ)$ for $0 ≤ θ < π$

$r = \frac { 3 } { \sqrt { \operatorname { cos } ( \theta ) \cdot \operatorname { sin } ( \theta ) } }$ for $0 ≤ θ < 2π$

9-118.

Pilar recognizes the graph from part (a) of problem 9-117 as $y = 4$ in the rectangular coordinate system.

If $r = \frac { 4 } { \operatorname { sin } ( \theta ) }$ is same as $y = 4$ , then write an equation for $y$ in terms of $r$ and $θ$ .
Confirm that $y = r \sin(θ)$ geometrically by using the diagram at right. Then, predict how an equation for $x$ can be written in terms of $r$ and $θ$ and verify that relationship using the diagram.
Write an equation that relates $r$ , $x$ , and $y$ . Justify your identity using the diagram.

Ray labeled, r, in center of first quadrant, starting at origin, point at end of ray labeled as ordered pair, (r, comma theta), vertical segment from point to positive x axis, labeled y, creating right triangle, with horizontal leg labeled, x, and central angle labeled theta.

9-119.

Pilar wants to convert more polar equations into rectangular form and vice versa. Use the identities below to help her rewrite the given equations. Simplify your equations and avoid square roots.

$\mathbf{x = r \cos(θ)}$ $\mathbf{y = r \sin(θ)}$ $\mathbf{x^2+y^2=r^2}$

$x^2 + y^2 = 9$

$r = 2\sin(θ)$

$r =\frac { 3 } { \sqrt { \operatorname { cos } ( \theta ) \cdot \operatorname { sin } ( \theta ) } }$

$r^2 = 2\sin(2θ)$

9-120.

Use the graph of $y = f^\prime(x)$ at right to complete parts (a) through (e) below. Homework Help ✎

For what $x$ -value(s) does $f$ have a relative minimum?
Where does $f$ have points of inflection?
Describe what happens on the graph of $y = f^{\prime\prime}(x)$ when $x = 5$ .
Describe the graph of $y = f(x)$ when $x = 0$ .
If $f(0) = 2$ , estimate $f(3)$ .

Continuous curve labeled, f prime of x, coming through the point (negative 1, comma 1), turning at the origin, changing from concave up to concave down @ (1, comma 0.25), turning @ (2, comma 0.5), changing concavity @ (4.5, comma negative 0.5), turning at (5, comma negative 2), passing through (6.25, comma 0), continuing up & right.

9-121.

Solve the differential equation $\frac { d y } { d x } = (1 + y^2)e^{2x}$ if $(0, 1)$ lies on the solution curve. Homework Help ✎

9-122.

For each of the following equations, what is $\frac { d y } { d x }$ ? Homework Help ✎

$y \ln(x) = x^2$

$\frac { d ^ { 2 } y } { d x ^ { 2 } }= 3x^2 − 2x$

$( \frac { x } { 2 } ) ^ { 2 } + ( \frac { y } { 3 } ) ^ { 2 }= 1$

$y =| 2 x |$

9-123.

Thoroughly investigate the graph of $f(x) = 3x^{1/3} - 4x$ . Identify all of the important qualities, such as where the function is increasing, decreasing, concave up, and concave down. Also identify point(s) of inflection and extrema. Be sure to justify all statements both graphically and analytically. Homework Help ✎

9-124.

A car is traveling north at $30$ miles per hour while a truck is traveling east at $40$ miles per hour. If both vehicles were located at the same place at $t = 0$ , how fast is the distance between them changing when $t = 3$ hours? Homework Help ✎

9-125.

Suppose that $x =\frac { 1 } { t ^ 2 + 1 }$ and $y = t^2$ . Homework Help ✎

Express $x$ as a function of $y$ .
Express $y$ as a function of $x$ .
How will the graph of the parametric equations given above be different if $x =\frac { 1 } { u + 1 }$ and $y = u$ ?

9-126.

On your polar graph paper, complete tables and plot points for $r_1 = 4\cos(θ)$ and $r_2 = 4\cos(2θ)$ . How does the “ $2θ$ ” change the graph? Homework Help ✎

9-127.

Timmy is tired. He does not want to add the infinite terms of $\frac { 1 } { 2 } + \frac { 1 } { 4 } + \frac { 1 } { 8 } + \frac { 1 } { 16 } + \ldots$ . So instead, he just adds the first three terms. Homework Help ✎

How far is his result from the actual sum of the infinite series? (This is called his “error.”)
What would his error have been if he added the first four terms?
Generalize his error. That is, if he adds up $n$ terms of this geometric series, what will his error be?

Calculus Third Edition

9.4.2What is the polar equation for a rose?

Polar Curves