Vectors

1. Given f(x,y,z)=x^2y^3z^6, in what direction is F(x,y,z) increasing most rapidly at the point P(1,-1,1). What is the rate of increase? 2. Locate and classify the critical points of the function h(x,y) = x^2 -4x+4xy+y^2-16y.

Differential Equations

Differential Equations. See attached file for full problem description. Problem #2 only.

Formulation of PDE's

1. Formulate a boundary value problem modeling heat conduction in a thin bar of length , if the left end is kept at temperature zero and the right end is insulated. The initial temperature in the cross section at is . 2. Formulate a boundary value problem for the motion of an elastic string of length , fastened at both ends and released from rest with an initial position given by .The string vibrates in the plane. Its motion is opposed by air resistance, which has a force at each point of magnitude proportional to the square of the velocity at that point. See attached file for full problem description.

Characteristics Curves

1. (a) Solve the given equation by the method of characteristic curves. (b) Check your answer by plugging it back into the equation. See attached file for full problem description.

Differential Equations

Differential Equations. See attached file for full problem description.