By using term by term integration find a series - find

Question

Question 1. Let f (x) = 1/(1 + 4x²)

a) Find McLaurin series of f (x) by using the series converging to 1/(1 - x)

b) By using term by term integration find a series converging to

1/2tan^-1 (2/3) = ₀∫^1/2f (x)dx

c) Find an approximated value for 1/2tan^-1 (2/3) with an error less than 0.0002. (Hint. Use the theorem for error estimation for approximating alternating series)

Question 2.

a) Consider the function

f (x, y) = {(5x² + 3y²)/(4x² + y²)     if (x, y) ≠ (0, 0),

              {               0                        if (x,y) = (0,0).

By using the definition of partial derivative find f_x(0, 0), f_x(0,k) and (f_x)_y(0, 0).

b) Show that the limit

lim         4xy²/(x² + 2y⁴)
(x,y)→(0,0)

does not exists by using (i) two-path test, (ii) polar coordinates.

Question 3.

a) Evaluate the integral ∫∫_R f(x, y)dA, where f(x, y) = 4y/(3 + 5x⁵) and R is the region in the first quadrant bounded by the curves y = x² and x=2.

b) Reverse the order of the integral of ₀∫^2√In3_y/2∫^√In3 e^5x2 dxdy (Do not evaluate it).

c) Evaluate the integral of _-4∫⁴ _-√(4-y²)∫^√(4-y2) ln(y² + x² + 1)dxdy (Hint. Change it into an equivalent polar integral).

d) Show that the integral _c∫(cos xy² + y)dx + (2y sin x + x)dy is path independent. Then evaluate it over a curve starting from the origin to (Π, 2).