Calculus Answers

if y=e^ax cos^3 x sin^2 x , find dy/dx.

Find the arclength of y=3x^(3/2) on 1<x<2

Consider the following integral:

Function is 1/9x
Lower limit= 0
Upper limit= 1

Determine whether the improper integral diverges or converges.
Evaluate the integral. (If the integral diverges, enter INFINITY. Round to two decimal places if the integral converges.)

a)what is the largest interval (a,b) where the Power series sigma from n=1 to infinity (x-5)^n/(2^(2n)root(n+1)) converges absolutely? [use the Ratio Test]
b) what is the radius of convergence of this Power series sigma from n=1 to infinity (x-5)^n/(2^(2n)root(n+1))?
c)Does the series converge (conditionally) at the endpoints of the interval?

Determine convergence or divergence of the following series [ Explain why. ]
a) sigma from n=0 to infinity 1/root(n^3+2)
b) sigma from n=0 to infinity (3^(n+1) 2^(2n))/(4^(2n-1))
c) sigma from n=2 to infinity 1/(root(n)+sin(n))

Let f(x ) = e^x cos(x ) . The power series expansion of f(x) centered at 0 is f(x) = 1 + x - x^3 /3 - x^4 /6 + c5 x^5 + x^7 / 630 + ...
a) show that c5 = - 1/30
b) estimate integral from 0 to 1 e^x cos(x) dx using the given series.
c) estimate f'(1) using the given series

A calculator gives us integral from 0 to 1/2 1/(1+x^7) dx ≈ 0.4995
Use power series to find an answer which is more accurate, correct to at least 7 decimals. Show and explain your work and give an estimate of the error

Use power series to compute sin(26.132742). Show and explain your work and give
an estimate of the error. [ 26.132742 is a number … not in degrees, if you feel more
comfortable you can read it as 26.132742 radians. ]

Find derivatives of the functions
cot(sec x)*sinh(log x^2)+cos(x^3)
x^x
exp(tan x^(sin x))

If x((1+y)^1/2)+y((1+x)^1/2)=0, then prove that dy/dx= - (1/(1+x)^2)