Calculus Answers

1. Find a power series representation centered at x = 0 of the function x² / (4-x²) and determine its interval of convergence.
2. Determine all numbers x so that the series "\\displaystyle\\sum_{k=0}^\u221e" (ln x)^k converges
3. For which p is the series "\\displaystyle\\sum_{n=0}^\u221e" n/ (n²+1)^p convergent?

Thank you so much.

use long division to express the function x²-3x+7/x+2 as the sum of a polynomial function and a rational function of the form A/x+2, where a is constant

List the critical numbers of the following function in increasing order. Enter N in any blank that you don't need to use.

𝑓(𝜃)=18cos(𝜃)+9sin^2(𝜃), −𝜋≤𝜃≤𝜋

Find the critical points of the function f(x, y) = (x²+5y²)e^{1-2x^2-2y^2} and use the second

derivative test to classify them as local maxima, local minima, or saddle point

Find the domain and range of the following function given by

𝑓(𝑥)=√(3𝑥−5)(𝑥+4)𝑥3−16𝑥.

a) Let Φ(u, v) = (2 sin u cos v, 3 sin u sin v, cos u), for π/2 ≤ u, v ≤ π. Plot and describe the surface S parametrized by Φ.

b) Find the equation of the tangent plane to S at (u, v) = (3π/4, 3π/4).

c) Find the volume of the region in R3 bounded by the surface S and the coordinate planes.

Find a complete and singular integrals of 2xz − px2 − 2qxy + pq = 0

Show that ∫¹₀ (sin(1/x))/xⁿ 𝑑𝑥 ; 𝑥 > 0 convergence absolutely, if 𝑛 < 1.

1. Calculate the turning points of the function y=sin3t using differential calculus
2. Show which are maxima, minima or points of inflexion using the second derivative

Describe the graph of a one-to-one function and its inverse?