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

Solution:

Given, r = e^θ

We have to find the points on the given curve where the tangent line is horizontal or vertical.

Using polar coordinates,

x = r cos θ = e^θcos θ

y = r sin θ = e^θsin θ

Horizontal tangent occurs where dy/dx = 0

Vertical tangent occurs where dy/dx → ∞ ⇒ dx →0

Now, dy/dx = (dy/dθ) / (dx/dθ)

dy/dθ = e^θsin θ + e^θcos θ

dx/dθ = -e^θsin θ + e^θcos θ

dy/dx = (e^θsin θ + e^θcos θ) / (- e^θsin θ + e^θcos θ)

So horizontal tangent occurs where

(e^θsin θ + e^θcos θ) / (-e^θsin θ + e^θcos θ) = 0

e^θsin θ + e^θcos θ = 0

e^θ(sin θ + cos θ) = 0

sin θ + cos θ = 0

sin θ = -cos θ

tan θ = -1

So, θ = 3π/4 and 7π/4

Therefore, the points on the curve are \((e^{\frac{3\pi }{4}},\frac{3\pi }{4})\) and \((e^{\frac{7\pi }{4}},\frac{7\pi }{4})\)

Vertical tangent occurs where

(e^θsin θ + e^θcos θ) / (- e^θsin θ + e^θcos θ) = ∞

-e^θsin θ + e^θcos θ = 0

e^θ(-sin θ + cos θ) = 0

sin θ = cos θ

tan θ = 1

So, θ = π/4 and 5π/4

Therefore, the points on the curve are \((e^{\frac{\pi }{4}},\frac{\pi }{4})\) and \((e^{\frac{5\pi }{4}},\frac{5\pi }{4})\).

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

Summary:

The points on the given curve r = e^θ where the tangent line is horizontal or vertical are \((e^{\frac{3\pi }{4}},\frac{3\pi }{4})\) and \((e^{\frac{7\pi }{4}},\frac{7\pi }{4})\) and \((e^{\frac{\pi }{4}},\frac{\pi }{4})\) and \((e^{\frac{5\pi }{4}},\frac{5\pi }{4})\).