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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.

56. r = 2 + sin 3θ, θ = π/4

To determine

To find: The slope of the tangent line for the polar curve r=2+sin3θ at the point θ=π4 .

Explanation

Given:

The slope of the tangent line for the polar curve r=2+sin3θ .

Calculation:

Substitute (2+sin3θ) for r in equation x=rcosθ .

x=rcosθ=(2+sin3θ)cosθx=2+sin3θcosθ

Substitute (2+sin3θ) for r in equation y=rsinθ .

y=rsinθ=(2+sin3θ)sinθy=(2+sin3θ)sinθ

Differentiate the equation (x=2+sin3θcosθ) with respect to θ .

x=2+sin3θcosθdxdθ=(2+sin3θ)(sinθ)+cosθ(3cos3θ)

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 .

Substitute [(2+sin3θ)(cosθ)+sinθ(3cos3θ)] for dydθ and [(2+sin3θ)(sinθ)+cosθ(3cos3θ)] for dxdθ in the above equation.

dydx=[(2+sin3θ)(cosθ)+sinθ(3cos3θ)][(2+sin3θ)(sinθ)+cosθ(3cos3θ)]

Find the slope of the tangent line for the polar curve r=2+sin3θ at the point θ=π4

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