9.4.3How do parameters affect polar graphs?

Polar Families

9-128.

SKETCHING POLAR CURVES

Your team will be assigned a generic polar equation from the list below.

Your Task:

Part 1: With your team, use your graphing calculator to investigate the graph of your polar curve. Look for patterns as you change the parameters, $a$ and $b$ . Use these patterns to sketch accurate graphs without a graphing calculator or a table.

Part 2: Find the team that had the same type of polar curve. Your team will merge with this team to prepare a presentation. Include the following information in your presentation:

Describe the general shape you investigated.
Describe the affect cosine and sine have on the position of the polar graph.
What happens when $θ = 0$ ?
At what values of $θ$ will the graph reach the pole, $(0, 0)$ ?
What minimum interval of $θ$ will create a complete graph?
How does changing the parameter, $a$ , affect the shape of the graph?
How does changing the parameter, $b$ , affect the shape of the graph?

Circles:

$r = a\cos(θ)$

$r = a\sin(θ)$

Cardioids:

$r = a + a\cos(θ)$

$r = a + a\sin(θ)$

Limaçons:

$r = b + a\cos(θ)$

$r = b + a\sin(θ)$

Roses:

$r = a\cos(bθ)$

$r = a\sin(bθ)$

Review and Preview problems below

9-129.

Given the graph of $y = f^\prime(x)$ below, draw a possible graph of $y = f(x)$ . Why is there more than one possible solution? Homework Help ✎

Continuous curve, coming from left just above x axis, changing from concave up to concave down at the point (0, comma 0.5), continuing to the right just below y = 1.

9-130.

If $$100$ has been earning continuously compounded interest at a rate of $5\%$ per year, what will the balance be after $25$ years? Homework Help ✎

9-131.

What is $\frac { d } { d x } \int _ { 4 } ^ { 6 x ^ { 2 } - 2 } ( 9 \sqrt { t - 1 } ) d t$ ? Homework Help ✎

9-132.

No calculator! Determine the dimensions of the rectangle with its base on the $x$ -axis and vertices on $y = 6 − 2x^2$ that has a maximum area. Homework Help ✎

$\$

9-133.

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 ✎

$\int \frac { 1 } { y ( 2 - y ) } d y$

$\int \operatorname { sec } ( m ) \operatorname { tan } ( m ) \operatorname { ln } ( \operatorname { sec } ( m ) )$

$\int \frac { 3 } { x ^ { 2 } + 4 x + 3 } d x$

$\int _ { 1 } ^ { \infty } \frac { 1 } { x ^ { 2 } } d x$

9-134.

No calculator! Calculate the average value of $y = 5x^3$ over $[ - \frac { \pi } { 4 } , \frac { \pi } { 4 } ]$ using two different methods. For each method, explain your process. Homework Help ✎

$\$

9-135.

For each of the series below, write an equivalent expression using sigma notation. Homework Help ✎

$5 + 10 + 20 + 40 + ... + 5(2)^{n-1}$

$1+\frac { 2 } { 3 } + \frac { 4 } { 9 } + \frac { 8 } { 27 } + \ldots$

9-136.

For any three points $P$ , $Q$ , and $R$ , express $\overrightarrow { Q R }$ in terms of $\overrightarrow { P Q }$ and $\overrightarrow { P R }$ . Support your answer with a diagram. Homework Help ✎

9-137.

The graph below is called a limaçon. Use your conclusions from problem 9-128 to write its equation. Homework Help ✎

Calculus Third Edition