Question

Question 8 Select the curve generated by the polar equation: r=sin(20) Then find the area enclosed by one petal & Q Q B Q Area: • Question 9 Write the power series representation of the following function and find the interval of convergence of the power series (in interval notation) f(0) = 27 6 + 73 00 f(x) = n=0 Interval of Convergence:

Answer 1

Question 8

T= sin(20)

8

The petal in first quadrant is traced as varies from 0 to 2

-{2 Area = 2dᎾ

1 sin? (20)do 0 2

1 + cos(40) -de 2 0 2

1 cos(40) 4 16

$=\frac{\pi}{8}$

Question 9

2 +9 = (2)!

1+ 6 6

6 + + +...(-1)" 61 6 6 62 63

3η Σ Σ(-1)». 6 6η n=0

3η+7 =Σ(-1) 61+1 n=0

For interval of convergence, applying Ratio Test: L = lim anti Jan 1 n00

lim 100 <1 6

- 76 75 76

67/3 Checking for L=1: At x = -76, an = (-1)"+3n+7! > 1, hence diverges 6

67/3 At x = V6, an = (-1)"4 =(-1)" > 1, hence diverges 6

Interval of convergence : - 76<< < 76