Q 49. - Calculus

Question

Graph the limac¸ons $r = 3 + \cos \theta , r = 3 + 3 \cos \theta , r = 3+4 \cos \theta , r = 3 + 6 \cos \theta$ Make a conjecture about the behavior of graphs of limac¸ons of the form $r = 3 + b \cos \theta$ for various values of $b$ In particular, try to understand which values of b will give a limac¸on with an inner loop. Which values of $b$ will give a limac¸on with a dimple? Which values of b will give a result in a convex

limac¸on?

Step-by-Step Solution

Answer

The graph is a convex one when $0\le b\le \frac{3}{2}$ The graph is a dimple when $\frac{3}{2}<b\le 3$

Step 1: Given information

$r = 3 + \cos \theta , r = 3 + 3 \cos \theta , r = 3+4 \cos \theta , r = 3 + 6 \cos \theta$

Step 2: Calculation

Consider the limacon $r=3+\cos \theta$

The goal is to figure out how graphs of the pattern $r=3+b\cos \theta$ behave.

Find different $r$ values by substituting different $\theta$ values.

Consider $\theta =0,\frac{\pi }{2},\pi ,\frac{3\pi }{2},2\pi$

For different $\theta$ values, we find the values of the equation $r=3+\cos \theta$

When $\theta =0$

$$\begin{gathered} r=3+\cos 0[\text{ since }r=3+\cos \theta ] \\ r=3+1 \text{ since }\cos 0=1] \\ r=4 \end{gathered}$$

Then the coordinate $(r,\theta )=(4,0)$

When $\theta =\frac{\pi }{2}$

$$\begin{gathered} r=3+\cos \frac{\pi }{2}[\text{ since }r=3+\cos \theta ] \\ r=3+0\left[\text{ since }\cos \frac{\pi }{2}=0\right] \\ r=3 \end{gathered}$$

Then the coordinate $(r,\theta )=\left(3,\frac{\pi }{2}\right)$

When $\theta =\pi$

$$\begin{gathered} r=3+\cos \pi [\text{ since }r=3+\cos \theta ] \\ r=3+(-1)\left[\text{ since }\cos \frac{\pi }{2}=0\right] \\ r=2 \end{gathered}$$

Then the coordinate $(r,\theta )=(2,\pi )$

When $\theta =\frac{3\pi }{2}$

Open with -

$$\begin{gathered} r=3+\cos \frac{3\pi }{2}[\text{ since }r=3+\cos \theta ] \\ r=3+0\left[\text{ since }\cos \frac{3\pi }{2}=0\right] \\ r=3 \end{gathered}$$

When $\theta =2\pi$

$$\begin{gathered} r=3+\cos 2\pi [\text{ since }r=3+\cos \theta ] \\ r=3+1[\text{ since }\cos 2\pi =1] \\ r=4 \end{gathered}$$

Then the coordinates are $(r,\theta )=(4,2\pi )$

We have distinct coordinates for different $\theta$ values.

Draw the graph by putting all of the following points on it.

Similarly, we can find the values of $r$ $r\cos r=3+4\cos \theta$ and $r=3+6\cos \theta$

Step 3: Calculation

Now plot the points on the graph and draw the different graphs. 

The inner loop grows in size as the $b$ value increases in the graph. The equation $r=3+\cos \theta$ is a cardioid. when $0\le b\le \frac{3}{2}$ the graph is convex. When $\frac{3}{2}<b\le 3$ the graph is a dimple.

Hence this is the explanation.