Then create and name your own curve

Sketch the curve and check for $x,y,$ and $r$ symmetry.

$r=\cos 3\theta$ (three petals)

10 $r^2=10+6\cos 4\theta$

11 $r=e^{\theta }$ (logarithmic spiral)

12 $r=1/\theta$ (hyperbolic spiral)

13 $r=\tan \theta$

14 $r=1-2\sin 3\theta$ (rose inside rose)

15 Convert $r=6\sin \theta +8\cos \theta$ to the $xy$ equation of a circle (what radius, what center ?).

*16 Squaring and adding in the Mars-Earth equation gives $x_{\mathrm{M}-E}^2+y_{\mathrm{M}-E}^2=13-12\cos \pi t$. The graph of $r^2=13-12\cos \theta$ is not at all like Figure 9.4d. What went wrong ?

In 17–23 find the points where the two curves meet.

17 $r=2\cos \theta$ and $r=1+\cos \theta$

Warning: You might set $2\cos \theta =1+\cos \theta$ to find $\cos \theta =1$. But the graphs have another meeting point—they reach it at different $\theta$’s. Draw graphs to find all meeting points.

18 $r^2=\sin 2\theta$ and $r^2=\cos 2\theta$

19 $r=1+\cos \theta$ and $r=1-\sin \theta$

20 $r=1+\cos \theta$ and $r=1-\cos \theta$

21 $r=2$ and $r=4\sin 2\theta$

22 $r^2=4\cos \theta$ and $r=1-\cos \theta$

23 $r\sin \theta =1$ and $r\cos (\theta -\pi /4)=\sqrt{2}$ (straight lines)

24 When is there a dimple in $r=1+b\cos \theta$ ? From $x=(1+b\cos \theta )\cos \theta$ find $dx/d\theta$ and $d^{2}x/d{\theta }^2$ at $\theta =\pi$. When that second derivative is negative the limaçon has a dimple.

25 How many petals for $r=\cos 5\theta$ ? For $r=\cos \theta$ there was one, for $r=\cos 2\theta$ there were four.

26 Explain why $r=\cos 100\theta$ has 200 petals but $r=\cos 101\theta$ only has 101. The other 101 petals are ____ . What about $r=\cos \frac{1}{2}\theta$ ?

27 Find an $xy$ equation for the cardioid $r=1+\cos \theta$.

28 (a) The flower $r=\cos 2\theta$ is symmetric across the $x$ and $y$ axes. Does that make it symmetric about the origin ? (Do two symmetries imply the third, so $-r=\cos 2\theta$ produces the same curve ?)

(b) How can $r=1$, $\theta =\pi /2$ lie on the curve but fail to satisfy the equation ?

29 Find an $xy$ equation for the flower $r=\cos 2\theta$.

30 Find equations for curves with these properties:

(a) Symmetric about the origin but not the $x$ axis

(b) Symmetric across the $45^{\circ }$ line but not symmetric in $x$ or $y$ or $r$

(c) Symmetric in $x$ and $y$ and $r$ (like the flower) but changed when $x\leftrightarrow y$ (not symmetric across the $45^{\circ }$ line).

Problems 31–37 are about conic sections—especially ellipses.

31 Find the top point of the ellipse in Figure 9.5a, by maximizing $y=r\sin \theta =\sin \theta /(1+\frac{1}{2}\cos \theta )$.

32 (a) Show that all conics $r=1/(1+e\cos \theta )$ go through $x=0$, $y=1$.

(b) Find the second focus of the ellipse and hyperbola. For the parabola ($e=1$) where is the second focus ?

33 The point $Q$ in Figure 9.5c has $y=1$. By symmetry find $x$ and then $r$ (negative!). Check that $x^2+y^2=r^2$ and $|QF|=2|Qd|$.

34 The equations $r=A/(1+e\cos \theta )$ and $r=1/(C+D\cos \theta )$ are the same if $C=$ ____ and $D=$ ____ . For the mirror image across the $y$ axis replace $\theta$ by ____ . This gives $r=1/(C-D\cos \theta )$ as in Figure 12.10 for a planet around the sun.

35 The ellipse $r=A/(1+e\cos \theta )$ has length $2a$ on the $x$ axis. Add $r$ at $\theta =0$ to $r$ at $\theta =\pi$ to prove that $a=A/(1-e^2)$. The Earth’s orbit has $a=92,600,000$ miles = one astronomical unit (AU).

36 The maximum height $b$ occurs when $y=r\sin \theta =A\sin \theta /(1+e\cos \theta )$ has $dy/d\theta =0$. Show that $b=y_{\max }=A/\sqrt{1-e^2}$.

37 Combine $a$ and $b$ from Problems 35–36 to find $c=\sqrt{a^2-b^2}=Ae/(1-e^2)$. Then the eccentricity $e$ is $c/a$. Halley’s comet is an ellipse with $a=18.1$ AU and $b=4.6$ AU so $e=$ ____ .

Comets have large eccentricity, planets have much smaller $e$ : Mercury .21, Venus .01, Earth .02, Mars .09, Jupiter .05, Saturn .05, Uranus .05, Neptune .01, Pluto .25, Kohoutek .9999.

38 If you have a computer with software to do polar graphs, start with these:

1. Flowers $r=A+\cos n\theta$ for $n=\frac{1}{2},3,7,8$; $A=0,1,2$

2. Petals $r=(\cos m\theta +4\cos n\theta )/\cos \theta$, $(m,n)=(5,3)$, $(3,5)$, $(9,1)$, $(2,3)$

3. Logarithmic spiral $r=e^{\theta /2\pi }$

4. Nephroid $r=1+2\sin \frac{1}{2}\theta$ from the bottom of a teacup

5. Dr. Fay’s butterfly $r=e^{\cos \theta }-2\cos 4\theta +{\sin }^5(\theta /12)$

Then create and name your own curve.

This exercise builds on another exercise in its source book, which isn't shown here.

Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) · §9.2 Polar Equations and Graphs, #9