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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
BuyFindarrow_forward

Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

Sketch the polar curve.

14. r = 2 cos(θ/2)

To determine

To sketch: The polar curve r=2cosθ2.

Explanation

Given:

The Polar equation for the variable r is as follows.

r=2cosθ2 (1)

To draw the curve for the values of θ and r:

Determine the value of r.

Substitute the value of 0° for θ.

r=(2×cos(0.5×0°)×π180)=2

Similarly substitute till 1080° for θ.

Summarize the values of θ and r as shown in the table (1).

θ r
0.00 2.00
10.00 1.99
20.00 1.97
30.00 1.93
40.00 1.88
50.00 1.81
60.00 1.73
70.00 1.64
80.00 1.53
90.00 1.41
100.00 1.29
110.00 1.15
120.00 1.00
130.00 0.85
140.00 0.68
150.00 0.52
160.00 0.35
170.00 0.17
180.00 0.00
190.00 -0.17
200.00 -0.35
210.00 -0.52
220.00 -0.68
230.00 -0.85
240.00 -1.00
250.00 -1.15
260.00 -1.29
270.00 -1.41
280.00 -1.53
290.00 -1.64
300.00 -1.73
310.00 -1.81
320.00 -1.88
330.00 -1.93
340.00 -1.97
350.00 -1.99
360.00 -2.00
370.00 -1.99
380.00 -1.97
390.00 -1.93
400.00 -1.88
410.00 -1.81
420.00 -1.73
430.00 -1.64
440.00 -1.53
450.00 -1.41
460.00 -1.29
470.00 -1.15
480.00 -1.00
490.00 -0.85
500.00 -0.68
510.00 -0.52
520.00 -0.35
530.00 -0.17
540.00 0.00
550.00 0.17
560.00 0.35
570.00 0.52
580.00 0.68
590.00 0.85
600.00 1.00
610.00 1.15
620.00 1.29
630.00 1.41
640.00 1.53
650.00 1.64
660.00 1.73
670.00 1.81
680.00 1.88
690.00 1.93
700.00 1.97
710.00 1.99
720.00 2.00
730.00 1.99
740.00 1.97
750.00 1.93
760.00 1.88
770.00 1.81
780.00 1.73
790.00 1.64
800.00 1.53
810.00 1.41
820.00 1.29
830.00 1.15
840.00 1.00
850.00 0.85
860.00 0.68
870.00 0.52
880.00 0.35
890.00 0.17
900.00 0.00
910.00 -0.17
920.00 -0.35
930.00 -0.52
940.00 -0.68
950.00 -0.85
960.00 -1.00
970.00 -1.15
980.00 -1.29
990.00 -1.41
1000.00 -1.53
1010.00 -1.64
1020.00 -1.73
1030.00 -1.81
1040.00 -1.88
1050.00 -1.93
1060.00 -1.97
1070.00 -1.99
1080.00 -2.00

Graph:

Draw the polar curve for the values of θ and r as shown in the figure (1) from table 1

To draw the curve for the values of x and y.

Determine the value of x:

x=rcosθ

Substitute the value of 1 for r and 0° for θ.

x=2×cos0°=2

Determine the value of y.

y=rsinθ

Substitute the value of 1 for r and 0° for θ.

y=2×sin0°=0

Summarize the values of θ,r,x and y as shown in the table (2).

θ r x y
0.00 2.00 2

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Sect-10.1 P-11ESect-10.1 P-12ESect-10.1 P-13ESect-10.1 P-14ESect-10.1 P-15ESect-10.1 P-16ESect-10.1 P-17ESect-10.1 P-18ESect-10.1 P-19ESect-10.1 P-20ESect-10.1 P-21ESect-10.1 P-22ESect-10.1 P-23ESect-10.1 P-24ESect-10.1 P-25ESect-10.1 P-26ESect-10.1 P-27ESect-10.1 P-28ESect-10.1 P-29ESect-10.1 P-30ESect-10.1 P-31ESect-10.1 P-32ESect-10.1 P-33ESect-10.1 P-34ESect-10.1 P-35ESect-10.1 P-36ESect-10.1 P-37ESect-10.1 P-38ESect-10.1 P-39ESect-10.1 P-40ESect-10.1 P-41ESect-10.1 P-42ESect-10.1 P-43ESect-10.1 P-44ESect-10.1 P-45ESect-10.1 P-46ESect-10.1 P-47ESect-10.1 P-48ESect-10.1 P-49ESect-10.1 P-50ESect-10.1 P-51ESect-10.1 P-52ESect-10.2 P-1ESect-10.2 P-2ESect-10.2 P-3ESect-10.2 P-4ESect-10.2 P-5ESect-10.2 P-6ESect-10.2 P-7ESect-10.2 P-8ESect-10.2 P-9ESect-10.2 P-10ESect-10.2 P-11ESect-10.2 P-12ESect-10.2 P-13ESect-10.2 P-14ESect-10.2 P-15ESect-10.2 P-16ESect-10.2 P-17ESect-10.2 P-18ESect-10.2 P-19ESect-10.2 P-20ESect-10.2 P-21ESect-10.2 P-22ESect-10.2 P-23ESect-10.2 P-24ESect-10.2 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P-52RECh-10 P-53RECh-10 P-54RECh-10 P-55RECh-10 P-56RECh-10 P-57RECh-10 P-58RECh-10 P-1PCh-10 P-2PCh-10 P-3PCh-10 P-4PCh-10 P-5PCh-10 P-6P

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