1Evaluate the improper integral from 1 to infinity of x^2 * e^(-x^3) dx.
2Consider the series S1 = sum(sin(1/(n^2 + n))), S2 = sum(n! / (n^3 * 4^n)), and S3 = sum(1 / (2^n + n)). Which of the following statements are correct?
3Find the area of the region bounded by the functions y = -x^2 + 2x and y = -x + 2.
4A rod has density delta(x) = 1 / (x^2 + 1) for 0 <= x <= 1. What is the x-coordinate of the center of mass of the rod?
5For a mixing problem in an 800 m^3 garage where 0.0035 m^3 of CO enters per minute and the mixture is removed at a rate of 0.08 m^3/min, what is the equilibrium volume of CO in the garage?
6Using Euler's method with step-size h = 0.1 for the initial-value problem y' = (2t - y)^2, y(0) = 2, what is the approximate value of y(0.1)?
7Identify the interval of convergence for the power series sum from n=1 to infinity of [(-1)^n * (x - 1)^n] / [2^n * sqrt(n)].
8If the Maclaurin series for f(x) = x^3 / (1 - x)^2 is sum(n * x^(n+2)) from n=1 to infinity, what is the value of the fifth derivative f^(5)(0)?
9Which of the following are the partial derivatives of z = 2xy - x^2 + 2y?
10Find the equation of the tangent plane to the surface z = 2xy - x^2 + 2y at the point (2, 1).
