2. find the slope of the tangent line to the polar curve r = sin(2θ) at θ = π/4. sketch a graph of the curve…

Explanation:

Step1: Recall the formula for the slope of a tangent in polar - coordinates

The formula for the slope of the tangent line to a polar curve $r = f(\theta)$ is $\frac{dy}{dx}=\frac{\frac{dr}{d\theta}\sin\theta + r\cos\theta}{\frac{dr}{d\theta}\cos\theta - r\sin\theta}$.

Step2: Differentiate $r=\sin(2\theta)$ with respect to $\theta$

Using the chain - rule, if $u = 2\theta$, then $\frac{dr}{d\theta}=\frac{d}{d\theta}\sin(2\theta)=2\cos(2\theta)$.

Step3: Evaluate $r$ and $\frac{dr}{d\theta}$ at $\theta=\frac{\pi}{4}$

When $\theta=\frac{\pi}{4}$, $r=\sin(2\times\frac{\pi}{4})=\sin(\frac{\pi}{2}) = 1$. When $\theta=\frac{\pi}{4}$, $\frac{dr}{d\theta}=2\cos(2\times\frac{\pi}{4})=2\cos(\frac{\pi}{2}) = 0$.

Step4: Substitute into the slope formula

Substitute $r = 1$, $\frac{dr}{d\theta}=0$, and $\theta=\frac{\pi}{4}$ into $\frac{dy}{dx}=\frac{\frac{dr}{d\theta}\sin\theta + r\cos\theta}{\frac{dr}{d\theta}\cos\theta - r\sin\theta}$. We get $\frac{dy}{dx}=\frac{0\times\sin(\frac{\pi}{4})+1\times\cos(\frac{\pi}{4})}{0\times\cos(\frac{\pi}{4})-1\times\sin(\frac{\pi}{4})}=\frac{\frac{\sqrt{2}}{2}}{-\frac{\sqrt{2}}{2}}=- 1$.

Answer:

The slope of the tangent line to the polar curve $r = \sin(2\theta)$ at $\theta=\frac{\pi}{4}$ is $-1$.

For the sketch:

1. The polar curve $r = \sin(2\theta)$ is a four - petaled rose. The general form of a rose curve is $r = a\sin(n\theta)$ or $r=a\cos(n\theta)$. When $n = 2$ (even), the number of petals is $2n=4$.
2. To sketch the tangent line at $\theta=\frac{\pi}{4}$, we know that in polar coordinates, the point on the curve is $(r,\theta)=(1,\frac{\pi}{4})$. In Cartesian coordinates, $x = r\cos\theta=1\times\cos(\frac{\pi}{4})=\frac{\sqrt{2}}{2}$ and $y = r\sin\theta=1\times\sin(\frac{\pi}{4})=\frac{\sqrt{2}}{2}$. The slope of the tangent line is $-1$. So the tangent line has the equation $y-\frac{\sqrt{2}}{2}=-(x - \frac{\sqrt{2}}{2})$ or $y=-x+\sqrt{2}$ in Cartesian coordinates. We can plot the four - petaled rose curve $r = \sin(2\theta)$ and the tangent line $y=-x+\sqrt{2}$ on the same set of axes.