1. Let f(x) = 1 + x^3/ 1 − x^3.
(a) Identify the intervals on which f is increasing;
(b) Identify the intervals on which f is decreasing;
(c) Identify any asymptotes of f;
(d) Find and classify the critical points of f;
(e) Find the points of inflection of f, if any.
2. An object of weight W is dragged along a horizontal plane by a force acting along a rope attached to the object. If the rope makes an angle θ with the plane, then the magnitude of the force is
F = µW / µ sin θ + cos θ
where µ is a constant called the coefficient of friction. For what value of θ is F smallest?
Compute each of (a) AB, (b) A^2, (c) B^2, (d) BA, or explain why it is not defined. (Hint: See Stroud Chapter 5 on Matrices.)
4. Given the following system of linear equations
3x + y + z = 4
2x − y − z = 6
x − 4y + 3z = −11
(a) Set up the matrix equation Ax = b;
(b) Find det(A);
(c) Find A−1;
(d) Use A−1 to solve the system.
(For part (c), there are two ways to do this. Method 1 is the so-called AI method, using Gauss-Jordan: see pages 251-255 of Week 12 notes. Method 2 uses the adjoint and determinant: see Stroud pages 499-506. You can choose whichever method you prefer.)
5. Perform the following integrals: