Calculus Answers

(D − 1)

2

(D

2 + 1)

2y = sin2

(

x

2

) + e

x + x

Determine whether the following series converges or diverges. If the series converges, find its sum.

a. ∑∞𝑛=1 (1+2ⁿ⁺¹)/3n

b. ∑∞𝑛=1 (3+2ⁿ)/(2ⁿ⁺²)

c. ∑∞𝑛=1 1/(4n²-1)

Find the value of a and b, if

Lim_x→∞ [x(1 + acosx) - b sin x]/x³ = 1

(7y=4t^2)dy+4tydt=0

Find fx(x,y), fy(x,y), fx(1,3), and fy(-2,4) for the given function. If

𝑧 = 𝑓(𝑥, 𝑦) = 3𝑥^3𝑦^2 − 𝑥^2𝑦^3 + 4𝑥 + 9

Let A be a singular row reduced echelon square matrix of size n .Find a matr xA(x)=(a_(ij)(x))_(n times n) where a_(ij)(x)' s are polynomials in x such that A(0)= (a_(ij)(0))_(n times n)=A but A(delta)=(a_(ij)(delta))_(n times n) is nonsingular for all delta!=0.

Solve the integral of x²cosxdx, using integration by parts.

Find fx(x,y), fy(x,y), fx(1,3), and fy(-2,4) for the given function. If 

𝑧 = 𝑓(𝑥, 𝑦) = 3𝑥ଷ𝑦ଶ − 𝑥ଶ𝑦ଷ + 4𝑥 + 9

) If 𝐼𝑛 = ∫ 𝑐𝑜𝑠𝑛𝑥 cos 𝑛𝑥 𝑑𝑥

𝜋

2

then find the reduction formula 

connecting 𝐼𝑛 and 𝐼𝑛−1 and hence prove that 𝐼𝑛 =

𝜋

2

𝑛+1

Evaluate _C∫ (2x² + 3y²)dx, where C is the curve given by x(t)=at² , y(t)=2at, 0≤t≤1.