A Collection of Problems in Differential Calculus: Problems from Calculus I Final Examinations, 2000-2020 Department of Mathematics, Simon Fraser University

Section 4.3 Polar Coordinates

1. Express the polar equation $r=\cos 2\theta$ in rectangular coordinates.

2. Sketch polar graphs of:

   1. $r=1+\sin \theta\text{.}$

   2. $r=\cos 3\theta\text{.}$

3. For the each of the following circles find a polar equation, i.e. an equation in $r$ and $\theta\text{:}$

   1. $\displaystyle x^2+y^2=4$

   2. $\displaystyle (x-1)^2+y^2=1$

   3. $\displaystyle x^2+(y-0.5)^2=0.25$

4. Find the maximum height above the $x$-axis of the cardioid $r=1+\cos \theta\text{.}$

5. Sketch the graph of the curve whose equation in polar coordinates is $r=1-2\cos\theta\text{,}$ $0\leq \theta \lt 2\pi\text{.}$

6. Sketch the graph of the curve whose equation in polar coordinates is $r=3\cos 3\theta\text{.}$

7. Sketch the curve whose polar equation is $r=-1+\cos \theta\text{,}$ indicating any symmetries. Mark on your sketch the polar coordinates of all points where the curve intersects the polar axis.

8. Sketch a polar coordinate plot of:

   1. $\displaystyle \ds r=\frac{1}{2}+\sin \theta$

   2. $\displaystyle r=2\cos 3\theta$

   3. $\displaystyle r^2=-4\sin 2\theta$

   4. $\displaystyle r=2\sin \theta$

   5. $\displaystyle r=2\cos \theta$

   6. $\displaystyle r=4+7\cos \theta$

9. Consider the curve given by the polar equation $r=1-\cos \theta\text{,}$ for $0\leq \theta \lt 2\pi\text{.}$

   1. Given a point $P$ on this curve with polar coordinates $(r,\theta)\text{,}$ represent its Cartesian coordinates $(x,y)$ in terms of $\theta\text{.}$

   2. Find the slope of the tangent line to the curve where $\ds \theta = \frac{\pi }{2}\text{.}$

   3. Find the points on this curve where the tangent line is horizontal or vertical.

10. Consider the curve given by the polar equation $r=\cos (2\theta)\text{,}$ for $0\leq \theta \lt 2\pi\text{.}$

    1. Find $\ds \frac{dy}{dx}$ in terms of $\theta\text{.}$

    2. Find the Cartesian coordinates for the point on the curve corresponding to $\ds \theta = \frac{\pi }{8}\text{.}$

    3. Find the tangent line to the curve at the point corresponding to $\ds \theta = \frac{\pi }{8}\text{.}$

    4. Sketch this curve for $\displaystyle 0\leq \theta \leq \frac{\pi}{4}$ and label the point from part (b) on your curve.

11. Consider the curve given by the polar equation $r=4\cos (3\theta)\text{,}$ for $0\leq \theta \lt 2\pi\text{.}$

    1. Find the Cartesian coordinates for the point on the curve corresponding to $\ds \theta = \frac{\pi }{3}\text{.}$

    2. One of graphs in Figure 4.3.1 is the graph of $r=4\cos(3\theta)\text{.}$ Indicate which one by circling it.

    3. Find the slope of the tangent line to the curve where $\ds \theta = \frac{\pi }{3}\text{.}$

12. Consider the curve given by the polar equation $r=4\sin (3\theta)\text{,}$ for $0\leq \theta \lt 2\pi\text{.}$

    1. Find the Cartesian coordinates for the point on the curve corresponding to $\ds \theta = \frac{\pi }{6}\text{.}$

    2. One of graphs in Figure 4.3.1 is the graph of $r=4\sin(3\theta)\text{.}$ Indicate which one by circling it.

    3. Find the slope of the tangent line to the curve where $\ds \theta = \frac{\pi }{3}\text{.}$

    

    

    

13. Consider the curve given by the polar equation $r=1+3\cos(2\theta)\text{,}$ for $0\leq \theta \lt 2\pi\text{.}$

    1. Find the Cartesian coordinates for the point on the curve corresponding to $\ds \theta = \frac{\pi }{6}\text{.}$

    2. One of graphs in Figure 4.3.2 is the graph of $r=1+3\cos(2\theta)\text{.}$ Indicate which one by putting a checkmark in the box below the graph you chose.

    3. Find the slope of the tangent line to the curve where $\ds \theta = \frac{\pi }{6}\text{.}$

    

    

    

14. Consider the curve given by the polar equation $r=1-2\sin \theta\text{,}$ for $0\leq \theta \lt 2\pi\text{.}$

    1. Find the Cartesian coordinates for the point on the curve corresponding to $\ds \theta = \frac{3\pi }{2}\text{.}$

    2. The curve intersects the $x$-axis at two points other than the pole. Find polar coordinates for these other points.

    3. On Figure 4.3.3 identify the graphs that correspond to the following two polar curves.

       $$\begin{equation*} \begin{array}{cc} \fbox { } \ r=1-2\sin \theta \amp \fbox { } \ r=1+2\sin \theta \end{array} \end{equation*}$$

       

       

       

15. Consider the curve $C$ given by the polar equation $r=1+2\cos \theta\text{,}$ for $0\leq \theta \lt 2\pi\text{.}$

    1. Find the Cartesian coordinates for the point on the curve corresponding to $\ds \theta = \frac{\pi }{3}\text{.}$

    2. Find the slope of the tangent line where $\ds \theta = \frac{\pi }{3}\text{.}$

    3. On Figure 4.3.4 identify the graph of $C\text{.}$

    

    

    

16. 1. Sketch a polar coordinate plot of

       $$\begin{equation*} r=1+2\sin 3\theta, \ 0\leq \theta \leq 2\pi\text{.} \end{equation*}$$

    2. How many points lie in the intersection of the two polar graphs

       $$\begin{equation*} r=1+2\sin 3\theta, \ 0\leq \theta \leq 2\pi \end{equation*}$$

       and

       $$\begin{equation*} r=1? \end{equation*}$$

    3. Algebraically find all values of $\theta$ that

       $$\begin{equation*} 1=1+2\sin 3\theta, \ 0\leq \theta \leq 2\pi\text{.} \end{equation*}$$

    4. Explain in a sentence or two why the answer to part (b) differs from (or is the same as) the number of solutions you found in part (c).

17. Consider the following curve $C$ given in polar coordinates as

    $$\begin{equation*} r(\theta )=1+\sin \theta +e^{\sin \theta }, \ 0\leq \theta \leq 2\pi\text{.} \end{equation*}$$

    1. Calculate the value of $r(\theta )$ for $\ds \theta =0, \frac{\pi }{2}, \frac{3\pi }{2}\text{.}$

    2. Sketch a graph of $C\text{.}$

    3. What is the minimum distance from a point on the curve $C$ to the origin? (i.e. determine the minimum of $|r(\theta )|=r(\theta )=1+\sin \theta +e^{\sin \theta }$ for $\theta \in [0,2\pi ]$)

18. 1. Give polar coordinates for each of the points $A\text{,}$ $B\text{,}$ $C$ and $D$ on Figure 4.3.5.

       

       

       

       

    2. On Figure 4.3.6 identify the graphs that correspond to the following three polar curves.

       $$\begin{equation*} \fbox { } \ r=1-2\sin \theta \ \ \fbox { } \ r^2\theta =1 \ \ \fbox { } \ r=\frac{1}{1-2\sin \theta} \end{equation*}$$

       

       

       

       

19. 1. Sketch the curve defined by $r=1+2\sin \theta\text{.}$

    2. For what values of $\theta\text{,}$ $\theta \in [-\pi ,\pi )\text{,}$ is the radius $r$ positive?

    3. For what values of $\theta\text{,}$ $\theta \in [-\pi ,\pi )\text{,}$ is the radius $r$ maximum and for what values is it minimum?

20. 1. Sketch the graph described in polar coordinates by the equation $r=\theta$ where $-\pi \leq \theta \leq 3\pi\text{.}$

    2. Find the slope of this curve when $\ds \theta =\frac{5\pi }{2}\text{.}$ Simplify your answer for full credit.

    3. Express the polar equation $r=\theta$ in cartesian coordinates, as an equation in $x$ and $y\text{.}$

21. 1. Let $C$ denote the graph of the polar equation $r=5\sin \theta\text{.}$ Find the rectangular coordinates of the point on $C$ corresponding to $\ds \theta =\frac{3\pi }{2}\text{.}$

    2. Write a rectangular equation (i.e. using the variables $x$ and $y$) for $C\text{.}$ (in other words, convert the equation for $C$ into rectangular coordinates.)

    3. Rewrite the equation of $C$ in parametric form, i.e. express both $x$ and $y$ as functions of $\theta\text{.}$

    4. Find an expression for $\ds \frac{dy}{dx}$ in terms of $\theta\text{.}$

    5. Find the equation of the tangent line to $C$ at the point corresponding to $\ds \theta =\frac{\pi }{6}\text{.}$

22. Find the slope of the tangent line to the polar curve $r=2$ at the points where it intersects the polar curve $r=4\cos \theta\text{.}$ (Hint: After you find the intersection points, convert one of the curves to a pair of parametric equations with $\theta$ as the perimeter.

23. A bee goes out from its hive in a spiral path given in polar coordinates by $r=be^{kt}$ and $\theta =ct\text{,}$ where $b\text{,}$ $k\text{,}$ and $c$ are positive constants. Show that the angle between the bee's velocity and acceleration remains constant as the bee moves outward.