[Math] Find the points on the given curve where the tangent line is horizontal or vertical.

calculuspolar coordinates

Please help! I don't know how else to do this question. Thank you!!

10

Best Answer

You are given $r$ as a function of $\theta$,i.e., $r=f(\theta)$, so you rewrite the usual identities: $$x = rcos(\theta)$$ $$y=rsin(\theta)$$

As: $$x=f(\theta)cos(\theta)$$ $$y=f(\theta)sin(\theta)$$

Then, it is a simple exercise to show that:$$\frac{dy}{dx}=\frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}} = \frac{f'(\theta)sin(\theta)+f(\theta)cos(\theta)}{f'(\theta)cos(\theta)-f(\theta)sin(\theta)}$$

Finally, since $f'(\theta)=\frac{d}{d\theta}e^{\theta}=e^{\theta}=f(\theta)$ this simplifies to:$$\frac{dy}{dx}=\frac{sin(\theta)+cos(\theta)}{cos(\theta)-sin(\theta)}$$

So, when is this final equation equal to $0$? When is it indeterminate? And what does this imply about the slope of the tangent line?

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