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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

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

63. r = 1 + cos θ

To determine

To find: The tangent line is horizontal or vertical with the point for the curve following relation r=1+cosθ .

Explanation

Given:

The polar curve is as below.

r=1+cosθ .

Calculation:

Horizontal tangents is obtained when the condition dydθ=0 is true, and the vertical tangents is obtained with the condition dxdθ=0 .

Differentiate the curve equation (r=1+cosθ) with respect to θ .

r=1+cosθdrdθ=sinθ

Write the chain rule for the condition dydx .

dydx=dydθdxdθ = drdθsinθ+rcosθdrdθcosθrsinθ

For horizontal tangents, the condition dydθ=0 is true.

Substitute (sinθ) for (drdθ) and (1+cosθ) for (r) in equation (drdθcosθrsinθ) .

dxdθ=drdθcosθrsinθ=(sinθ)cosθ(1+cosθ)sinθ=sinθcosθsinθcosθsinθ=2sinθcosθsinθ=sinθ(2cosθ+1)

Substitute (sinθ) for (drdθ) and (1+cosθ) for (r) in equation (drdθsinθ+rcosθ) .

dydθ=drdθsinθ+rcosθ

dydθ=(sinθ)sinθ+(1+cosθ)cosθ=sin2θ+cosθ+cos2θ=cos2θ+cosθ=2cos2θ1+cosθ=(2cosθ1)(cosθ+1)

Horizontal tangents is obtained by the condition dydθ=0 .

Substitute (2cosθ1)(cosθ+1) for (dydθ) .

dydθ=0(2cosθ1)(cosθ+1)=0cosθ=12 or 1

θ=π3,π,5π3

Substitute (π3) for θ in equation r=1+cosθ .

r=1+cosθ=1+cos(π3)=32

Substitute (π) for θ in equation r=1+cosθ .

r=1+cosθ=1cos(π)=0

Substitute (5π3) for θ in equation r=1+cosθ

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