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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

Find the slope of the tangent line to the given polar curve at the point specified by the value of θ.

57. r = 1/θ, θ = π

To determine

To find: The slope of the tangent line for the polar curve r=1θ at the point θ=π .

Explanation

Given:

The slope of the tangent line for the polar curve r=1θ .

Calculation:

Differentiate the curve equation r with respect to θ .

r=1θdrdθ=1θ2

Differentiate the parametric equation (y=(2+sin3θ)sinθ) with respect to θ .

y=(2+sin3θ)sinθdydθ=(2+sin3θ)(cosθ)+sinθ(3cos3θ)

Write the chain rule for dydx .

dydx=dydθdxdθ = drdθsinθ+rcosθdrdθcosθrsinθ

Substitute (1θ2) for (drdθ) and (1θ) for (r) in equation (drdθsinθ+rcosθdrdθcosθrsinθ)

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