7.8: Slopes and Curve Sketching in Polar Coordinates
- Page ID
- 155879
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Derivatives can be used to measure direction in polar as well as in rectangular coordinates. We begin with two theorems, one about the direction of a curve at the origin (an unusual point in polar coordinates) and the other about the direction of a curve elsewhere. Then we shall use these theorems for sketching curves.
At any value \(\theta_{0}\) where the curve \(r=f(\theta)\) passes through the origin, the curve is tangent to the line \(\theta=\theta_{0}\).
More precisely; if \(r=0\) at \(\theta=\theta_{0}\) but \(r \neq 0\) for all \(\theta \neq \theta_{0}\) in some neighborhood of \(\theta_{0}\), then
\[\lim _{\theta \rightarrow \theta_{0}} \frac{\Delta y}{\Delta x}=\tan \theta_{0}, \quad \lim _{\theta \rightarrow \theta_{0}} \frac{\Delta x}{\Delta y}=\cot \theta_{0} \nonumber \]
PROOF Suppose \(\cos \theta_{0} \neq 0\), so \(\tan \theta_{0}\) exists. Let \(\triangle \theta\) be a nonzero infinitesimal. Then
\(\Delta r \neq 0\) and \(r\) changes from 0 to \(\Delta r\). We compute \(\Delta y / \Delta x\).
\[\begin{aligned} \Delta y & =(0+\Delta r) \sin \left(\theta_{0}+\Delta \theta\right)-0 \sin \theta_{0} \\ & =\Delta r \sin \left(\theta_{0}+\Delta \theta\right) \\ \Delta x & =\Delta r \cos \left(\theta_{0}+\Delta \theta\right) \\ \frac{\Delta y}{\Delta x} & =\frac{\Delta r \sin \left(\theta_{0}+\Delta \theta\right)}{\Delta r \cos \left(\theta_{0}+\Delta \theta\right)}=\tan \left(\theta_{0}+\Delta \theta\right) \end{aligned} \nonumber \]
Taking standard parts,
\[\lim _{\theta \rightarrow 0_{0}} \frac{\Delta y}{\Delta x}=\tan \theta_{0} \nonumber \]
Similarly, when \(\sin \theta_{0} \neq 0\),
\[\lim _{\theta \rightarrow \theta_{0}} \frac{\Delta x}{\Delta y}=\cot \theta_{0} \nonumber \]
Both limits were given in the theorem to cover the case where the curve is vertical and \(\tan \theta_{0}\) is undefined.
The theorem tells us that if \(r=0\) at \(\theta_{0}\), the curve must approach the origin from the \(\theta_{0}\) direction. Figure 7.8 .1 shows two cases.
(a) If \(r\) has a local maximum or minimum at \(\theta_{0}\), then \(r\) has the same sign on both sides of \(\theta_{0}\). In this case the curve has a cusp at \(\theta_{0}\).
(b) If \(r\) has no local maximum or minimum at \(\theta_{0}\), then \(r\) is positive on one side of \(\theta_{0}\) and negative on the other side. In this case the curve crosses the origin at \(\theta_{0}\).
(a)
We now consider points other than the origin. In rectangular coordinates, the slope of a curve \(y=f(x)\) at a point \(P\) is \(d y / d x=\tan \phi\) where \(\phi\) is the angle between the \(x\)-axis and the line tangent to the curve at \(P\) as shown in Figure 7.8.2.
Figure 7.8.2
When \(r \neq 0\) in polar coordinates, a useful measure of the direction of the curve at a point \(P\) is \(\tan \psi\), where \(\psi\) is the angle between the radius \(O P\) and the tangent line at \(P\) (see Figure 7.8.3).
The following theorem gives a simple formula for \(\tan \psi\) when \(r \neq 0\).
Suppose \(r=f(\theta)\) is a curve in polar coordinates and dr/d日 exists at a point \(P\) where \(r \neq 0\). Let \(L\) be the line tangent to the curve at \(P\) and let \(\psi\) be the angle between \(O P\) and \(L\). Then
\[\begin{aligned} & \cot \psi=\frac{1}{r} \frac{d r}{d \theta} \\ & \text { If } d r / d \theta \neq 0 \\ & \tan \psi=\frac{r}{d r / d \theta} \end{aligned} \nonumber \]
DISCUSSION When \(r=0, P\) is the origin so the line \(O P\) and angle \(\psi\) are undefined.
Figure 7.8 .4
The formula can be seen intuitively in Figure 7.8.4.
\(\Delta \theta\) is infinitesimal. As we move from the point \(P(r, \theta)\) to the point \(Q(r+\Delta r, \theta+\Delta \theta)\) on the curve, the change in the direction perpendicular to \(O P\) will be very close to \(\Delta \theta\), so we have
\[\frac{\Delta r}{r \Delta \theta} \approx \cot \psi, \quad \frac{1}{r} \frac{d r}{d \theta}=\cot \psi \nonumber \]
We shall postpone the proof to the end of this section.
We can use Theorem 2 in curve sketching as follows.
(a) In an interval where \(\tan \psi>0\), the curve is going away from the origin as \(\theta\) increases because \(d r / d \theta\) has the same sign as \(r\).
(b) Where \(\tan \psi<0\), the curve is going toward the origin as \(\theta\) increases because \(d r / d \theta\) has the opposite sign as \(r\).
(c) Where \(r\) has either a local maximum or minimum and \(d r / d \theta\) exists, the curve is going in a direction perpendicular to the radius. This is because \(d r / d \theta=0\) so \(\cot \psi=0\).
Each of these cases is shown in Figure 7.8.5.
Polar coordinates are best suited for trigonometric functions, which have the property that \(f(\theta)=f(\theta+2 \pi)\). We shall therefore concentrate on the interval \(0 \leq \theta<2 \pi\).
Suppose that the function \(r=f(\theta)\) is differentiable for \(0 \leq \theta \leq 2 \pi\). The following steps may be used in sketching the curve.
(a)
(b)
(c)
Figure 7.8.5
Step 1 Compute \(d r / d \theta\).
Step 2 Find all points where \(r=0\) or \(d r / d \theta=0\).
Step 3 Sketch \(y=f(x)\) in rectangular coordinates. (A method for doing this is given in Section 3.9.)
Step 4 Compute \(r, d r / d \theta\), and \(\tan \psi=r(d r / d \theta)\) at the points where \(r=0\) or \(d r / d \theta=0\) and at least one point between. Make a table, and test for local maxima or minima.
Step 5 Draw a smooth curve using the rectangular graph of step three and the table of step four.
Sketch the curve \(r=1+\cos \theta\).
Solution
Step \(1 d r / d \theta=-\sin \theta\).
Step \(2 \quad r=0\) when \(\theta=\pi . \quad d r / d \theta=0\) when \(\theta=0, \pi\).
Step 3 See Figure 7.8.6.
Step 4
| \(\theta\) | \(r=1+\cos \theta\) | \(d r / d \theta\) | \(\tan \psi\) | Comments |
|---|---|---|---|---|
| 0 | 2 | 0 | - | \(\max\) |
| \(\pi / 2\) | 1 | -1 | -1 | \(|r|\) decreasing |
| \(\pi\) | 0 | 0 | - | min, cusp at 0 |
| \(3 \pi / 2\) | 1 | 1 | 1 | \(|r|\) increasing |
Figure 7.8.7
Step 5 We draw the curve in Figure 7.8.7. The curve is called a cardioid because of its heart shape.
Sketch the curve \(r=\sin 2 \theta\).
Solution
Step \(1 d r / d \theta=2 \cos 2 \theta\).
Step \(2 \quad r=0\) at \(\theta=0, \frac{\pi}{2}, \pi, \frac{3 \pi}{2} . \quad d r / d \theta=0\) at \(\theta=\frac{\pi}{4}, \frac{3 \pi}{4}, \frac{5 \pi}{4}, \frac{7 \pi}{4}\).
Step 3 See Figure 7.8.8.
Step 4 We take values at intervals of \(\frac{\pi}{8}\) beginning at \(\theta=0\). We can save some time by observing that the values from \(\pi\) to \(2 \pi\) are the same as those from 0 to \(\pi\).
| \(\theta\) | \(r=\sin 2 \theta\) | \(d r / \theta\) | \(\tan \psi\) | Comments | |
|---|---|---|---|---|---|
| 0 and \(\pi\) | 0 | 2 | 0 | crosses origin | |
| \(\pi / 8\) and \(9 \pi / 8\) | \(\sqrt{2} / 2\) | \(\sqrt{2}\) | \(1 / 2\) | \(|r|\) increasing | |
| \(2 \pi / 8\) and \(10 \pi / 8\) | 1 | 0 | - | max | |
| \(3 \pi / 8\) and \(11 \pi / 8\) | \(\sqrt{2} / 2\) | \(-\sqrt{2}\) | \(-1 / 2\) | \(|r|\) decreasing | |
| \(4 \pi / 8\) and \(12 \pi / 8\) | 0 | -2 | 0 | crosses origin | |
| \(5 \pi / 8\) and \(13 \pi / 8\) | \(-\sqrt{2} / 2\) | \(-\sqrt{2}\) | \(1 / 2\) | \(|r|\) increasing | |
| \(6 \pi / 8\) and \(14 \pi / 8\) | -1 | 0 | - | min | |
| \(7 \pi / 8\) and \(15 \pi / 8\) | \(-\sqrt{2} / 2\) | \(\sqrt{2}\) | \(-1 / 2\) | \(|r|\) decreasing |
Step 5 We plot the points and trace out the curve as \(\theta\) increases from 0 to \(2 \pi\). Figure 7.8 .9 shows the curve at various stages of development. The graph looks like a four-leaf clover.
If \(r\) approaches \(\infty\) as \(\theta\) approaches 0 or \(\pi\), the curve may have a horizontal
\(0 \leq \theta \leq \frac{\pi}{4}\)
\(0 \leq \theta \leq \frac{2 \pi}{4}\)
\(0 \leq \theta \leq \frac{5 \pi}{4}\)
\(0 \leq \theta \leq \frac{7 \pi}{4}\)
\(0 \leq \theta \leq 2 \pi\)
Figure 7.8.9
asymptote which can be found by computing the limit of \(y\). At \(\theta=\pi / 2\) or \(3 \pi / 2\) there may be vertical asymptotes. The method is illustrated in the following example.
Sketch \(r=\tan \left(\frac{1}{2} \theta\right)\).
Solution
Step \(1 d r / d \theta=\frac{1}{2} \sec ^{2}\left(\frac{1}{2} \theta\right)\).
\[\begin{aligned} y & =r \sin \theta=\sin \frac{1}{2} \theta \sin \theta / \cos \frac{1}{2} \theta \\ \cdot & =\sin \frac{1}{2} \theta\left(2 \sin \frac{1}{2} \theta \cos \frac{1}{2} \theta\right) / \cos \frac{1}{2} \theta=2 \sin ^{2}\left(\frac{1}{2} \theta\right) \end{aligned} \nonumber \]
Step \(2 \quad r=0\) at \(\theta=0\).
\(r\) is undefined at \(\theta=\pi\).
\(d r / d \theta\) is never 0 .
Step 3 See Figure 7.8.10.
Figure 7.8 .10
| tep 4 | \(r\) or \(\lim r\) | \(\lim y\) | \(d r / d \theta\) | \(\tan \psi\) | Comments |
|---|---|---|---|---|---|
| 0 | 0 | \(1 / 2\) | crosses origin | ||
| \(\pi / 2\) | 1 | 1 | 1 | \(|r|\) increasing | |
| \(\theta \rightarrow \pi^{-}\) | \(\infty\) | 2 | asymptote \(y=2\) | ||
| \(\theta \rightarrow \pi^{+}\) | \(-\infty\) | 2 | asymptote \(y=2\) | ||
| \(3 \pi / 2\) | -1 | 1 | -1 | \(|r|\) decreasing |
Step 5 The curve crosses itself at the point \(x=0, y=1\), because this point has both polar coordinates
\[(r=1, \theta=\pi / 2),(r=-1, \theta=3 \pi / 2) \nonumber \]
Figure 7.8.11 shows the graph for various stages of development.
PROOF OF THEOREM 2 Assume the curve is not vertical at the point \(P\), that is, \(d x \neq 0\). Since
\[x=r \cos \theta, \quad y=r \sin \theta \nonumber \]
we have
\[\frac{d y}{d x}=\frac{d y / d \theta}{d x / d \theta}=\frac{r \cos \theta+(d r / d \theta) \sin \theta}{-r \sin \theta+(d r d \theta) \cos \theta} \nonumber \]
By the definition of the tangent line \(L\) (see Figure 7.8.12),
\[\frac{d y}{d x}=\frac{\text { change in } y \text { along } L}{\text { change in } x \text { along } L}=\frac{\sin (\theta+\psi)}{\cos (\theta+\psi)} \nonumber \]
Using the addition formulas,
\[\frac{d y}{d x}=\frac{\sin \theta \cos \psi+\cos \theta \sin \psi}{\cos \theta \cos \psi-\sin \theta \sin \psi} \nonumber \]
Thus \(\frac{r \cos \theta+(d r / d \theta) \sin \theta}{-r \sin \theta+(d r / d \theta) \cos \theta}=\frac{\sin \psi \cos \theta+\cos \psi \sin \theta}{-\sin \psi \sin \theta+\cos \psi \cos \theta}\).
Multiplying out and canceling, we get
\[\begin{gathered} r \cos \psi\left(\sin ^{2} \theta+\cos ^{2} \theta\right)=\frac{d r}{d \theta} \sin \psi\left(\sin ^{2} \theta+\cos ^{2} \theta\right) \\ r \cos \psi=\frac{d r}{d \theta} \sin \psi, \quad \frac{1}{r} \frac{d r}{d \theta}=\cot \psi \end{gathered} \nonumber \]
If the curve is vertical at \(P\) we may use the same proof but with \(d x / d y\) instead of \(d y / d x\).
PROBLEMS FOR SECTION 7.8
In Problems 1-6, find \(\tan \psi\), where \(\psi\) is the angle between a line through the origin and the curve.
\[\begin{array}{llll} \mathbf{1} & r=\theta & \mathbf{2} & r=\sin \theta \\ \mathbf{3} & r=\cos \theta & \mathbf{4} & r=\sec \theta \\ \mathbf{5} & r=1+\cos \theta & \mathbf{6} & r=\sin (20) \end{array} \nonumber \]
In Problems 7-25, sketch the given curve in polar coordinates by the method described in the text; \(0 \leq \theta \leq 2 \pi\) unless stated otherwise.
\[\begin{aligned} & 7 \quad r=\sin \theta+\cos \theta \\ & 8 \quad r=2+2 \sin \theta \\ & 9 \quad r=1 \frac{1}{2}+\sin \theta \\ & 10 \quad r=2+\cos \theta \\ & 11 \quad r=\frac{1}{2}+\cos \theta \\ & 12 \quad r=\cos \left(\frac{1}{2} \theta\right), \quad 0 \leq \theta \leq 4 \pi \\ & 13 \quad r=\sin \left(\frac{1}{3} \theta\right), \quad 0 \leq 0 \leq 6 \pi \\ & 14 \quad r=\sin ^{2} \theta \end{aligned} \nonumber \]
| \(r=1+3 \cos ^{2}(2 \theta)\) | 16 | \(r=\sin ^{2}(3 \theta)\) |
|---|---|---|
| \(r=\tan \theta\) | 18 | \(r=\sec \left(\frac{1}{2} \theta\right), \quad 0<\theta<4 \pi\) |
| \(r=1+\sec \theta\) | 20 | \(r=\frac{1}{1-\cos \theta}\) |
| \(r=\frac{1}{1+\sin \theta}\) | 22 | \(r=\cot (2 \theta)\) |
| \(r=\pi / \theta, \quad 0<\theta<\infty\) | 24 | \(r=1+\pi / 0, \quad 0<\theta<\infty\) |
In Problems 26-29, find the points where \(x\) and \(y\) have maxima and minima.
\(26 \quad r=1+\cos \theta\)
27
\[r=1+\sin ^{2} \theta \nonumber \]
\[r=\frac{3}{2}+\cos \theta \tag{29} \]
\(28 \quad r=\sin (20)\)
30 Find all points where the curves \(r=1+\cos \theta\) and \(r=3 \cos \theta\) intersect.
31 Find all points where the curves \(r=\frac{1}{2}\) and \(r=\sin (2 \theta)\) intersect. Warning: The points \((r, \theta)\) and \((-r, \pi+\theta)\) are the same.
32 Find all points where the curves \(r=\cos \theta\) and \(r=\sin (2 \theta)\) intersect.