Question

Find the points on the given curve where the tangent line is horizontal or vertical
r=e^(theta)

Answer 1

I think I did this right but I'm not sure.

So the point where the tangent line is horizontal means the slope or derivative is equal to zero. And where its vertical, the slope or derivative has to be undefined.

The derivative of a polar is \[dy/dx = (dr/d \theta * \sin \theta + r* \cos(\theta))/(dr/d \theta *\cos(\theta)-r*\sin(\theta))\]
so \[dr/d \theta = e^\theta\]
and \[r = e^\theta\]
So then you set
\[(e^\theta∗\sinθ+e^\theta∗\cos(θ))=0\]
to get the theta for horizontal slope
and
\[(e^\theta∗\cos(θ)−e^\theta∗\sin(θ) = 0\]
to get the theta for undefined slope.

Wolframalpha.com gives \[\theta = \pi * n - \pi/4\] for horizontal and
\[\theta = \pi*n - 3*\pi/4\] for vertical/undefined slope. n = any positive integer such as 1,2,3,4...

Hope this helps/is close to right.

Answer 2

Thank you!!

Answer 3

anytime :D