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Section 6.15 Polar Coordinates

These answers correspond to the problems in Section 4.3.

  1. (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

  2. See Figure 6.15.1.

    Figure 6.15.1. r=1+\sin \theta and r=\cos 3\theta
  3. (a) r = 2\text{,} (b) r = 2\cos \theta\text{,} (c) r = \sin \theta\text{.}

  4. 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
  5. See Figure 6.15.3.

    Figure 6.15.3. r=1-2\cos \theta
  6. See the graph r=\cos 3\theta (Figure 6.15.1) and apply the appropriate stretching.

  7. See Figure 6.15.2.

    Figure 6.15.4. \ds r=\frac{1}{2}+\sin \theta
  8. 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
  9. (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{.}

  10. (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
  11. (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
  12. (a) (2\sqrt{3},2)\text{.} (b) See Figure 6.15.8. (c) \ds \sqrt{3}\text{.}

  13. (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{.}

  14. (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{.}

  15. (a) (1,\sqrt{3})\text{.} (b) \ds \frac{1}{3\sqrt{3}}\text{.} (c) B.

  16. (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{.}

  17. (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{.}

  18. (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.

  19. (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
  20. (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
  21. (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{.}

  22. 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{.}

  23. 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