A solid has a circular base of radius 1 each cross-section

Question

Final Review-

Problem 1- Prove that- lim_x→1(x² + x - 3/x + 2) = -(1/3), using the ∈, δ definition of limit.

Problem 2- Show that the equation x⁴ - 10x² + 5 = 0 has a root in the interval (0, 2).

Problem 3- Let f(x) = 1/2-x. Find a general formula for f⁽ⁿ⁾(x).

Problem 4- Let f(x) = xe^x. Prove that f⁽ⁿ⁾(x) = (x + n)e^x.

Problem 5- Evaluate - lim_x→0(√(1 + tan x) - √(1 + sin x)/x³.

Problem 6- Suppose f and g are differentiable functions such that f(g(x)) = x and f'(x) = 1 + (f(x))². Show that g'(x) = 1/1+x².

Problem 7- Find two positive integers m, n such that m + 4n = 1000 and mn is as large as possible.

Problem 8- Find the point on the hyperbola xy = 8 that is closest to the point (3, 0).

Problem 9- If ₀∫⁴e^{(x-2)^4} dx = k, find the value of ₀∫⁴xe^{(x-2)^4} dx.

Problem 10- If f(x) = ₀∫^xx²sin(t²) dt, find f'(x).

Problem 11- Find the interval [a, b] for which the value of the integral _a∫^b(2 + x - x²) dx is a maximum.

Problem 12- A solid has a circular base of radius 1. Each cross-section of the solid by a plane perpendicular to the base is a square. Compute the volume of the solid.

Problem 13- Let {(x, y, z) : x² + y² = 1} and {(x, y, z) : x² + z² = 1} be two cylinders of radius 1. Find the volume of the solid defined by the intersection of these cylinders.

Problem 14- Let R be the region bounded by the curves y = 4x² - x³ and y = 0. Use cylindrical shells to find the volume of the solid obtained by rotating R about the y-axis.