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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

Use a graph to estimate the y-coordinate of the highest points on the curve r = sin 2θ. Then use calculus to find the exact value.

To determine

To find: The exact value of y -coordinates of the highest point on curve r=sin2θ .

Explanation

Given:

The polar curve equation r=sin2θ

Calculation:

The equation for the variable x is given below,

x=rcosθ (1)

The equation for the variable y is given below,

y=rsinθ (2)

The polar equations r=sin2θ .

Substitute (sin2θ) for r in equation (1),

x=rcosθ=(sin2θ)cosθ (3)

Substitute (sin2θ) for r in equation (2),

y=rsinθ=(sin2θ)sinθ (4)

Substitute (0) for θ in equation (3),

x=[(sin2θ)]cosθ=(sin2(0))cos(0)=0

Substitute (0) for θ in equation (4),

y=[(sin2θ)]sinθ=(sin2(0))sin(0)=0

The values of x and y for each value of θ is tabulated in the below table.

θ

(degrees)

θ

(rad)

x y
0 0 0 0
10 0.174611111 0.3369641 0.059443036
20 0.349222222 0.6042135 0.220022724
30 0.523833333 0.7501014 0.433305887
40 0.698444444 0.7542915 0.633327845

From the graph shown below in figure 1 the highest point of y -coordinates is 0

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