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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270336

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BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270336
Textbook Problem

What is the smallest viewing rectangle that contains every member of the family of polar curves r = 1 + c sin θ, where 0 ≤ c ≤ 1 ? Illustrate your answer by graphing several members of the family in this viewing rectangle.

To determine

To Find: The different rectangle curves from the given polar equation and smallest viewing rectangle that contains every member of the family of polar curves.

Explanation

Given:

The given polar curve is r=1+csinθ (1)

Calculation:

Calculate the value of x.

x=rcosθ

Substitute 1+csinθ for r in the above equation.

x=rcosθx=(1+csinθ)cosθ=cosθ+csinθcosθ

Substitute 12sin2θ for sinθcosθ in the above equation.

x=cosθ+csinθcosθx=cosθ+c(12sin2θ)

Calculate the value of y.

y=rsinθ

Substitute 1+csinθ for r in the above equation.

y=rsinθy=(1+csinθ)sinθy=sinθ+csin2θ

Provide different values for c and plot the graph.

Take the value of c=1

θ x=cosθ+1(12sin2θ) y=sinθ+(1)sin2θ
0 1.00 0.00
10 1.16 0.20
20 1.26 0.46
30 1.30 0.75
40 1.26 1.06
50 1.14 1.35
60 0.93 1.62
70 0.66 1.82
80 0.34 1.95
90 0.00 2.00
100 -0.34 1.95
110 -0.66 1.82
120 -0.93 1.62
130 -1.14 1.35
140 -1.26 1.06
150 -1.30 0.75
160 -1.26 0.46
170 -1.16 0.20
180 -1.00 0.00
190 -0.81 -0.14
200 -0.62 -0.23
210 -0.43 -0.25
220 -0.27 -0.23
230 -0.15 -0.18
240 -0.07 -0.12
250 -0.02 -0.06
260 0.00 -0.01
270 0.00 0.00
280 0.00 -0.01
290 0.02 -0.06
300 0.07 -0.12
310 0.15 -0.18
320 0.27 -0.23
330 0.43 -0.25
340 0.62 -0.23
350 0.81 -0.14
360 1.00 0.00

Take the value of c=23

θ x=cosθ+(23)(12sin2θ) y=sinθ+(23)sin2θ
0 1.00 0.00
10 1.10 0.19
20 1.15 0.42
30 1.15 0.67
40 1.09 0.92
50 0.97 1.16
60 0.79 1.37
70 0.56 1.53
80 0.29 1.63
90 0.00 1.67
100 -0.29 1.63
110 -0.56 1.53
120 -0.79 1.37
130 -0.97 1.16
140 -1.09 0.92
150 -1.15 0.67
160 -1.15 0.42
170 -1.10 0.19
180 -1.00 0.00
190 -0.87 -0.15
200 -0.73 -0.26
210 -0.58 -0.33
220 -0.44 -0.37
230 -0.31 -0.37
240 -0.21 -0.37
250 -0.13 -0.35
260 -0.06 -0.34
270 0.00 -0.33
280 0.06 -0.34
290 0.13 -0.35
300 0.21 -0.37
310 0.31 -0.37
320 0.44 -0.37
330 0.58 -0.33
340 0.73 -0.26
350 0.87 -0.15
360 1.00 0.00

Take the value of c=13

θ x=cosθ+(13)(12sin2θ) y=sinθ+(13)sin2θ
0 1.00 0.00
10 1.04 0.18
20 1.05 0.38
30 1.01 0.58
40 0.93 0.78
50 0.81 0.96
60 0.64 1.12
70 0.45 1.23
80 0.23 1.31
90 0.00 1.33
100 -0.23 1.31
110 -0.45 1.23
120 -0.64 1.12
130 -0.81 0.96
140 -0

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