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Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095

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BuyFindarrow_forward

Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095
Textbook Problem

Finding Slopes of Tangent Lines In Exercises 63 and 64, find dy / dx and the slopes of the tangent lines shown on the graph of the polar equation.

r = 2 ( 1 sin θ )

Chapter 10.4, Problem 60E, Finding Slopes of Tangent Lines In Exercises 63 and 64, find dy / dx and the slopes of the tangent

To determine

To Calculate:

The term dydx for equation r=2(1sinθ) and the slopes of tangent shown on graph of polar equation.

Explanation

Given:

The equation is r=2(1sinθ) and figure is shown as below:

Formula Used:

The formula is y=rsinθ

Calculation:

Consider the equation,

r=2(1sinθ)

Multiply the above equation both sides with sinθ,

rsinθ=2sinθ(1sinθ)

As it is known that  y=rsinθ substitute this value in above equation,

y=2sinθ2sin2θ

Now differentiate both sides with respect to θ

dydθ=2cosθ4sinθcosθ..(1)

In similarway multiply the given polar equation with cosθ,

rcosθ=2cosθ(1sinθ)

As it is known that x=rcosθ substitute this value in above equation,

x=2cosθ(1sinθ)

x=2cosθ2cosθsinθ

Now differentiate both sides with respect to θ,

dxdθ=2sinθ2cos2θ+2sin2θ

=22sinθ4cos

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