Calculus Answers

The functions f and g are defined by f(x)=1/|1-3x| and g(x)=log_1/3(1 /3x-2)-log₃(x) respectively

1.1. Write down the sets D_f(the domain of f) and D_g(the domain of g)

1.2. Solve the inequality f(x) greater than 2 for x"\\in" D_f

1.3. Solve the inequality g(x) greater or equals to 0 for x"\\in" D_g

Hint: use the change of base formula

A space probe in the shape of the sphere x ^ 2 + y ^ 2 + z ^ 2 = 30 enters Earth's atmosphere and its surface begins to heat. After 1 hour, the temperature at the point (z. y. :) on the probe's surface is T(x, y, z) = x - 2y + 5z Find the hottest point on the probe's surface.

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.