Question

Find the points on the given curve where the tangent line is horizontal or vertical

r=e^(theta)

Answer 1

r=e^θ using polar coordinates x=rcos(θ)=e^θcosθ
y=rsin(θ)=e^θsinθ

Horizontal tangents occur where dy/dx=0 and vertical tangents occur where

dy/dx→∞⟹dx→0.

dy/dx=dy/dθ/dx/dθ

dy/dx=e^θcosθ+e^θsinθ/−e^θsinθ+e^θcosθ

So horizontal tangents occur where e^θcosθ+e^θsinθ=0⟹θ=−π/4

Vertical tangents occur where −e^θsinθ+e^θcosθ=0⟹θ=π/4