Single Variable Calculus: Concepts and Contexts, Enhanced Edition
Single Variable Calculus: Concepts and Contexts, Enhanced Edition
4th Edition
ISBN: 9781337687805
Author: James Stewart
Publisher: Cengage Learning
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Chapter H.1, Problem 64E
To determine

To calculate:

The y coordinate value and the exact value of the curve r=sin2θ

Expert Solution & Answer
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Answer to Problem 64E

The y coordinate value of the curve r=sin2θ is 0.9428 .

Explanation of Solution

Given information:

  r=sin2θ

Calculation:

The given equation of the curve is:

  r=sin2θ

It is drawn as below:

  Single Variable Calculus: Concepts and Contexts, Enhanced Edition, Chapter H.1, Problem 64E

From the above graph:

The y coordinate value of highest points on the curve is 0.75 . Now, find the horizontal tangent line (dydθ=0) of the curve to estimate the y coordinate value of highest points on the curve r=sin2θ :

  y=rsinθy=sin2θsinθy=2sinθcosθsinθy=2sin2θcosθ

Take the derivate with respect to θ :

  y=2sin2θcosθdydθ=ddθ(2sin2θcosθ)dydθ=2sin2θ(sinθ)+22sinθcos2θdydθ=2sin3θ+4sinθcos2θdydθ=2sinθ(sin2θ2cos2θ)0=2sinθ(sin2θ2cos2θ)2sinθ(sin2θ2cos2θ)=0sinθ(sin2θ2(1sin2θ))=0sinθ(3sin2θ2)=0sinθ=0OR3sin2θ2=0θ=sin1(0)sinθ=23θ=0θ=sin123

Now put the calculated values of θ in the equation of the curve:

Case1: θ=0

  r=sin2θ=sin(2×0)=0

Case1: θ=sin123

  r=sin2θ=sin2(sin123)=0.9428

So, the y coordinate value and the exact value of the curve r=sin2θ is 0.9428 .

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