   Chapter 10.3, Problem 64E

Chapter
Section
Textbook Problem

Find the points on the given curve where the tangent line is horizontal or vertical.64. r = eθ

To determine

To find: The point for the curve r=eθ where the tangent line is horizontal or vertical.

Explanation

Given:

The polar curve is r=eθ.

Calculation:

Horizontal tangents is obtained when dydθ=0 and the vertical tangents is obtained by dxdθ=0.

Differentiate the curve equation (r=eθ) with respect to θ.

r=eθ

Substitute eθ for r in equation x=rcosθ,

x=rcosθ=(eθ)cosθ

Substitute eθ for r in equation y=rsinθ,

y=rsinθ=(eθ)sinθ

Differentiate the equation (x=(eθ)cosθ) with respect to θ.

dxdθ=eθsinθ+eθcosθ

Differentiate the equation (y=(eθ)sinθ) with respect to θ.

dydθ=eθcosθ+eθsinθ

Write the chain rule for dydx.

dydx=(dydθ)(dxdθ)=eθcosθ+eθsinθeθsinθ+eθcosθ=eθ(cosθ+sinθ)eθ(cosθsinθ)=(cosθ+sinθ)(cosθsinθ)

For horizontal tangents dydθ=0.

That is, (cosθ+sinθ)=0.

Square on both sides to get.

(cosθ+sinθ)2=0cos2θ+sin2θ+2sinθcosθ=01+sin2θ=0sin2θ=1

2θ=2nππ2θ=nππ4

Horizontal tangents is obtained by dydθ=0

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