Calculus Answers

Q.Choose the correct answer.
The radius of convergence for power series ∑_(n=1)^∞▒(2^k 〖(x+1)〗^(k+1))/k is
a.3/2
b.1
c.∞
d.½

Q. Which of the following function is periodic
a. sin x+sin √2x
b. sin 2x+cos 3x
c. e^xsin x
d. xsin x+cos x

Q. Solve ∫_(-∞)^∞▒x^2 e^(-x^2 )cos xdx.

Q. choose the correct option.
Q. Let H(x)=∫_a^x▒f(t)dt, for a≤x≤b,then which of the following is true?
(i) H may not be continuous on [a,b]
(ii) f is continuous at c∈[a,b],then H is differentiable at c.
(iii) H is continuous on [a,b].
a.(i) only
b.(i) and (ii) only
c.(ii) and (iii) only.
d.(iii) only

Q. choose the correct option.
Q. Which of the following statement is not true?
a. If f is bounded on [a,b], then f is Reimann integrable on [a,b].
b. If f is monotone on [a,b], then f is Reimann integrable on [a,b].
c. If f is continuous on [a,b], then f is Reimann integrable on [a,b].
d. If f is Lebesgue measurable on [a,b], then f is Reimann integrable on [a,b].

Q. choose the correct option.
Q. If f(x)= 1, 0<x≤1/2. -1, ½<x≤1 then
a. ∫_0^x▒f(s)ds=x, 0<x≤1/2. -x, ½<x≤1
b. ∫_0^x▒f(s)ds=x, 0<x≤1/2. 1-x, ½<x≤1
c. ∫_0^x▒f(s)ds=1/2, 0<x≤1/2. 1, ½<x≤1
d. ∫_0^x▒f(s)ds=1-x, 0<x≤1/2. x, ½<x≤1

First principle of differentiation

Q. Choose the correct answer.
Q. lim┬(n→∞)⁡□((a^n-a^(-n))/(a^n+a^(-n) )) =?
a. 0
b. 1
c. -1
d. ∞

Find the rate of change of h(x)=2 cos⁡〖(3x+tan⁡x)〗 with respect to x.

Differentiate the following:
5ln(2x−2y)−2y^2+5x=0