Section 6.15 Polar Coordinates
These answers correspond to the problems in Section 4.3.(x^2+y^2)^3=(y^2-x^2)\text{.} Multiply by r^2 and use the fact that \cos 2\theta =cos ^2\theta -\sin ^2\theta
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See Figure 6.15.1.
Figure 6.15.1. r=1+\sin \theta and r=\cos 3\theta (a) r = 2\text{,} (b) r = 2\cos \theta\text{,} (c) r = \sin \theta\text{.}
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On the given cardioid, x = (1 + \cos \theta ) \cos \theta and y = (1 + \cos \theta ) \sin \theta\text{.} The question is to find the maximum value of y\text{.} Note that y > 0 is equivalent to \sin \theta > 0\text{.} From \ds \frac{dy}{d\theta } = 2\cos ^2 \theta + \cos \theta -1 we get that the critical numbers of the function y = y(\theta ) are the values of \theta for which \ds \cos \theta = \frac{-1 \pm 3}{4}\text{.} It follows that the critical numbers are the values of \theta for which \ds \cos \theta = -1 or \ds \cos \theta = \frac{1}{2}\text{.} Since y_{\mbox{max} } > 0 it follows that \ds \sin \theta = \sqrt{1-\left( \frac{1}{2}\right) ^2}=\frac{\sqrt{3}}{2} and the maximum height equals \ds y = \frac{3\sqrt{3}}{4}\text{.} See Figure 6.15.2.
Figure 6.15.2. Curves r=1+\cos \theta and r=-1+\cos \theta and the points that correspond to \theta=0 -
See Figure 6.15.3.
Figure 6.15.3. r=1-2\cos \theta See the graph r=\cos 3\theta (Figure 6.15.1) and apply the appropriate stretching.
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See Figure 6.15.2.
Figure 6.15.4. \ds r=\frac{1}{2}+\sin \theta -
For (a) see Figure 6.15.4 and for (b) see Figure 6.15.1. For (c) and (d) see Figure 6.15.5 and for (e) and (f) see Figure 6.15.6.
Figure 6.15.5. r^2=-4\sin 2\theta and r=2\sin \theta Figure 6.15.6. r=2\cos \theta and r=4+7\cos \theta (a) (x,y)=((1-\cos\theta)\cdot \cos\theta, (1-\cos\theta)\cdot \sin\theta)\text{.} (b) \ds \left.\frac{dy}{dx}\right|_{\theta=\frac{\pi}{2}}=-1\text{;} y=-x+1\text{.} (c) Horizontal tangent lines at \ds \left(-\frac{3}{4},\frac{3\sqrt{3}}{4}\right), \ \left(-\frac{3}{4},-\frac{3\sqrt{3}}{4}\right)\text{;} Vertical tangent line at \ds (0,0), \ (-2,0), \left(\frac{1}{4},\frac{\sqrt{3}}{4}\right),\ \left(\frac{1}{4},-\frac{\sqrt{3}}{4}\right)\text{.}
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(a) \ds \frac{dy}{dx}=\frac{-2\sin 2\theta\sin\theta+\cos\theta\cos2\theta}{-2\sin 2\theta\cos\theta-\sin \theta\cos 2\theta}\text{.} (b) \ds \left(\frac{1}{2}\sqrt{1+\frac{1}{\sqrt{2}}},\frac{1}{2}\sqrt{1-\frac{1}{\sqrt{2}}}\right)\text{.} (c) y- \frac{1}{2}\sqrt{1-\frac{1}{\sqrt{2}}}= \frac{2\sqrt{2-\sqrt{2}}-\sqrt{2+\sqrt{2}}}{2\sqrt{2+\sqrt{2}}+\sqrt{2-\sqrt{2}}}\cdot\left(x-\frac{1}{2}\sqrt{1+\frac{1}{\sqrt{2}}}\right)\text{.} (d) See Figure 6.15.7.
Figure 6.15.7. Curve \ds r=\cos 2\theta, 0\leq \theta\leq \frac{\pi}{4} and a point -
(a) (-2,-2\sqrt{3})\text{.} (b) See Figure 6.15.8. (c) \ds -\frac{1}{\sqrt{3}}\text{.}
Figure 6.15.8. r=4\cos 3\theta and r=4\sin 3\theta (a) (2\sqrt{3},2)\text{.} (b) See Figure 6.15.8. (c) \ds \sqrt{3}\text{.}
(a) \left(\frac{5\sqrt{3}}{4},\frac{5}{4}\right)\text{.} (b) Observe that r(0)=r(\pi)=4 and \ds r\left(\frac{\pi}{2}\right)= r\left(\frac{3\pi}{2}\right)=1\text{.} (c) \ds \frac{\sqrt{3}}{23}\text{.}
(a) (0,-3)\text{.} (b) Solve y=(1-2\sin \theta)\sin \theta =0\text{.} (\pm 1,0)\text{.} (c) The middle graph corresponds to r=1+\sin 2\theta and the right graph corresponds to r=1-2\sin \theta\text{.}
(a) (1,\sqrt{3})\text{.} (b) \ds \frac{1}{3\sqrt{3}}\text{.} (c) B.
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(a) See Figure 6.15.9.
Figure 6.15.9. r=1+2\sin 3\theta, 0\leq \theta \leq 2\pi (b) 9. (c) \theta = 0\text{,} \ds \frac{\pi }{3}\text{,} \ds \frac{2\pi }{3}\text{,} \pi\text{,} \ds \frac{4\pi }{3}\text{,} \ds \frac{5\pi }{3}\text{,} 2\pi\text{.} (d) The remaining points of intersection are obtained by solving -1 = 1 + 2 \sin 3\theta\text{.}
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(a) r(0) = 2\text{,} \ds r \left( \frac{\pi }{2}\right) = 2+e\text{,} \ds r \left( \frac{3\pi }{2}\right) = e^{-1}\text{.} (b) See Figure 6.15.10.
Figure 6.15.10. r(\theta )=1+\sin \theta +e^{\sin \theta} (c) From \ds \frac{dr}{d\theta }= \cos \theta (1 + e^{\sin \theta }) = 0 we conclude that the critical numbers are \ds \frac{\pi }{2} and \ds \frac{3\pi }{2}\text{.} By the Extreme Value Theorem, the minimum distance equals e^{-1}\text{.}
(a) \ds A=\left( r=\sqrt{2},\theta= \frac{\pi }{4}\right)\text{,} \ds B=\left( 4,\frac{5\pi }{3}\right)\text{,} \ds C=\left( 2,\frac{7\pi }{6}\right)\text{,} \ds D=\left( 2\sqrt{2}-1,\frac{3\pi }{4}\right)\text{;} (b) A, B, D.
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(a) See Figure 6.15.11. (b) Solve \ds \sin \theta >-\frac{1}{2} to get \ds \theta \in \left[ -\pi, -\frac{5\pi }{6}\right) \cup \left( -\frac{\pi }{6},\pi \right)\text{.} (c) To find critical numbers solve \ds \frac{dr}{d\theta }=2\cos \theta =0 in [-\pi ,\pi )\text{.} It follows that \ds \theta =-\frac{\pi }{2} and \ds \theta =\frac{\pi }{2} are critical numbers. Compare r(-\pi )=r(\pi )=1\text{,} \ds r\left( -\frac{\pi }{2}\right) =-1\text{,} and \ds r\left( \frac{\pi }{2}\right) =3 to answer the question.
Figure 6.15.11. r(\theta )=1+2\sin \theta -
(a) See Figure 6.15.12. (b) The slope is given by \ds \left. \frac{dy}{dx}\right| _{\theta =\frac{5\pi }{2}}\text{.} From x=r\cos \theta =\theta \cos \theta and y=\theta \sin \theta it follows that \ds \frac{dy}{dx}=\frac{\frac{dy}{d\theta }}{\frac{dx}{d\theta }}=\frac{\sin \theta +\theta \cos \theta }{\cos \theta -\theta \sin \theta}\text{.} Thus \ds \left. \frac{dy}{dx}\right| _{\theta =\frac{5\pi }{2}}=-\frac{2}{5\pi }\text{.} (c) \ds \sqrt{x^2+y^2}=\arctan \frac{y}{x}\text{.}
Figure 6.15.12. r(\theta )=\theta\text{,} -\pi \leq \theta \leq 3\pi (a) (0,5)\text{.} (b) x^2+y^2=5y\text{.} (c) x=5\sin \theta \cos \theta\text{,} y=5\sin ^2\theta\text{.} (d) \ds \frac{dy}{dx}=\frac{2\sin \theta\cos \theta }{\cos ^2\theta - \sin ^2\theta}=\tan 2\theta\text{.} (e) \ds y -\frac{5}{4}=\sqrt{3}\left( x-\frac{5\sqrt{3}}{4}\right)\text{.}
Solve 2=4\cos \theta to get that curve intersect at ( 1, \sqrt{3})\text{.} To find the slope we note that the circle r=2 is given by parametric equations x=2\cos \theta and y=2\sin \theta\text{.} It follows that \ds \frac{dy}{dx}=\frac{2\cos \theta }{-2\sin \theta }=-\cot \theta\text{.} The slope of the tangent line at the intersection point equals \ds \left. \frac{dy}{dx}\right| _{\theta = \frac{\pi }{3}}=-\frac{\sqrt{3}}{3}\text{.}
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See Figure 6.15.13 for the graph of the case b=1\text{,} k=0.01\text{,} and c=2\text{.} The position in (x,y)-plane of the bee at time t is given by a vector function \ds \vec{s}(t)=\langle be^{kt}\cos ct,be^{kt}\sin ct\rangle\text{.} Recall that the angle \alpha between the velocity and acceleration is given by \ds \cos\alpha=\frac{\vec{v}\cdot\vec{a}}{|\vec{v}||\vec{a}|}\text{,} where \vec{v}(t)=\vec{s}^\prime(t) and \vec{a}(t)=\vec{s}^{\prime\prime}(t)\text{.} One way to solve this problem is to consider that the bee moves in the complex plane. In that case its position is given by
\begin{equation*} F(t)=be^{kt}\cos ct+i\cdot be^{kt}\sin ct=be^{kt}(\cos ct +i\sin ct)=be^{(k+ic)t}\text{,} \end{equation*}where i is the imaginary unit. Observe that \vec{v}(t)=\langle \mbox{Re} (F^\prime(t)),\mbox{Im} (F^\prime(t))\rangle and \vec{a}(t)=\langle \mbox{Re} (F^{\prime\prime}(t)),\mbox{Im} (F^{\prime\prime}(t))\rangle\text{.} Next, observe that F^\prime(t)=(k+ic)F(t) and F^{\prime\prime}(t)=(k+ic)^2F(t)\text{.} From F^{\prime\prime}(t)=(k+ic)F^\prime(t) it follows that \mbox{Re} (F^{\prime\prime}(t))=k\cdot \mbox{Re} (F^\prime(t))-c\cdot \mbox{Im} (F^\prime(t)) and \mbox{Im} (F^{\prime\prime}(t))=k\cdot \mbox{Im} (F^\prime(t))+c\cdot \mbox{Re} (F^\prime(t))\text{.} Finally, \vec{v}\cdot\vec{a}=\mbox{Re} (F^\prime(t))\cdot \mbox{Re} (F^{\prime\prime}(t))+\mbox{Im} (F^\prime(t))\cdot \mbox{Im} (F^{\prime\prime}(t))=k((\mbox{Re} (F^\prime(t)))^2+(\mbox{Im} (F^\prime(t)))^2)=k|F^\prime(t)|^2 which immediately implies the required result.
Figure 6.15.13. \ds r(t )=e^{0.01t}\text{,} \theta =2t\text{,} 0\leq t\leq 10\pi