-/6.76 points details my notes larcalc11 10.4.064. find dy/dx and the slopes of the tangent lines shown on…

Explanation:

Step1: Recall polar - to - rectangular conversion and derivative formula

We know that $x = r\cos\theta=(2 + 3\sin\theta)\cos\theta=2\cos\theta+3\sin\theta\cos\theta$ and $y = r\sin\theta=(2 + 3\sin\theta)\sin\theta=2\sin\theta+3\sin^{2}\theta$. The formula for $\frac{dy}{dx}=\frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}}$. First, find $\frac{dx}{d\theta}$ and $\frac{dy}{d\theta}$. $\frac{dx}{d\theta}=- 2\sin\theta+3(\cos^{2}\theta-\sin^{2}\theta)=-2\sin\theta + 3\cos2\theta$ (using the double - angle formula $\cos2\theta=\cos^{2}\theta-\sin^{2}\theta$). $\frac{dy}{d\theta}=2\cos\theta + 6\sin\theta\cos\theta=2\cos\theta+3\sin2\theta$ (using the double - angle formula $\sin2\theta = 2\sin\theta\cos\theta$). So, $\frac{dy}{dx}=\frac{2\cos\theta + 3\sin2\theta}{-2\sin\theta+3\cos2\theta}$.

Step2: Evaluate $\frac{dy}{dx}$ at $\theta=\frac{3\pi}{2}$

When $\theta=\frac{3\pi}{2}$, $\cos\theta = 0$, $\sin\theta=-1$, and $\cos2\theta=-1$, $\sin2\theta = 0$. $\frac{dy}{dx}\big|_{\theta=\frac{3\pi}{2}}=\frac{2\times0+3\times0}{-2\times(-1)+3\times(-1)}=\frac{0}{2 - 3}=0$.

Step3: Evaluate $\frac{dy}{dx}$ at $\theta=\pi$

When $\theta=\pi$, $\cos\theta=-1$, $\sin\theta = 0$, $\cos2\theta = 1$, $\sin2\theta = 0$. $\frac{dy}{dx}\big|_{\theta=\pi}=\frac{2\times(-1)+3\times0}{-2\times0+3\times1}=\frac{-2}{3}=-\frac{2}{3}$.

Step4: Evaluate $\frac{dy}{dx}$ at $\theta=\frac{\pi}{6}$

When $\theta=\frac{\pi}{6}$, $\cos\theta=\frac{\sqrt{3}}{2}$, $\sin\theta=\frac{1}{2}$, $\cos2\theta=\frac{1}{2}$, $\sin2\theta=\frac{\sqrt{3}}{2}$. $\frac{dy}{dx}=\frac{2\times\frac{\sqrt{3}}{2}+3\times\frac{\sqrt{3}}{2}}{-2\times\frac{1}{2}+3\times\frac{1}{2}}=\frac{\sqrt{3}+\frac{3\sqrt{3}}{2}}{-1 + \frac{3}{2}}=\frac{\frac{2\sqrt{3}+3\sqrt{3}}{2}}{\frac{1}{2}} = 5\sqrt{3}$.

Answer:

$\frac{dy}{dx}=\frac{2\cos\theta + 3\sin2\theta}{-2\sin\theta+3\cos2\theta}$ at $\left(-1,\frac{3\pi}{2}\right)$: $\frac{dy}{dx}=0$ at $(2,\pi)$: $\frac{dy}{dx}=-\frac{2}{3}$ at $\left(3.5,\frac{\pi}{6}\right)$: $\frac{dy}{dx}=5\sqrt{3}$