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

Question

Assume (0≤θ≤2π). Enter your answers as a comma-separated list of ordered pairs.)

r=e^(θ)





horizontal tangent	(r, θ) =
vertical tangent	(r, θ) =

Apparently I keep getting this answer wrong.

Answer 1

Here us a suggestion.

Since y = rsinθ, y = e^θsinθ. Similarly, x=e^θcosθ.

Find dy/dθ and dx/dθ. Then dy/dx = dy/dθ / dx/dθ.

Horizontal tangent lines when dy/dx = 0. Vertical tangent lines when dy/dx is undefined.

I get answers like this: Vertical: (e^π/4, π/4), (e^5pi/4, 5π/4).

Horizontal: (e^3π/4,3π/4), (e^7π/4, 7π/4).

If those answers appear to match what you are looking for and you need additional clarification, let me know--perhaps I can create a video answer with more detail.