Sample Final Exam

Problems from this sample final (given in Fall 2002) are assigned as homework problem as listed on the syllabus. (E.g., S4a on the syllabus refers to problem 4(a) on this final.)

1. Find all critical points of the function f (x, y) = x² + y² + xy² . Then classify each critical point as a local maximum, local minimum, or saddle point.

2. The temperature at the point (x, y, z) in space is given by T(x, y, z) = xyz³. T is measured in degrees Celsius ( ^oC); x, y, and z are measured in kilometers (km). At a certain instant of time, a space ship is at the point (2,3,1) and headed toward the point (3,4,3) at a rate of 5 kilometers per second (5 km/s). At what rate is the temperature experienced by the spaceship increasing at that instant? Your answer should be in terms of degrees Celsius per second ( ^oC/s).

3. (a) Let M be the surface parametrized by f(s, t) = (t, t², s³). Find the equation of the tangent plane to this surface at the point (2,4,1). Express your answer in the form Ax + By + Cz = D.

   (b) Find a parametrization for the plane in R³ with equation 4x - y = 4; that is, find an equation of the form x = x₀ + sa + tb.

4. Let S be the solid bounded above by the sphere x² + y² + z² = 16 and below by the paraboloid z = (x² + y²)/6.

   (a) Parametrize the solid S using cylindrical coordinates.

   (b) Let F(x, y, z) = (x + yz, sin(x⁹z⁶), cos(x⁷y⁸)). Use the Divergence Theorem to find the flux of F through the boundary of S. Use the outward pointing normal.

5. Let F(x, y, z) = (- y, x, sin(x⁷y⁸z⁹)) and let M be the surface given by z = x² + y² - 9, z < 0.

   (a) Give a parametrization of the boundary of M as a curve in 3-space.

   (b) Using any valid method, evaluate the integral of curl F over the surface M, where M has the outward pointing normal.

6. Let M be the surface x² + y² + z² = 4, z > 0. Using the outward-pointing normal, find the flux through M for the vector field F(x, y, z) = (y, x, z).

7. Let f(x, y) = (y², x + y) and let g be a function from R² to R such that the matrix of partial derivatives of g is given by [sin(x²), cos(xy)]. Let h be the composite h = gof.

   (a) Find the matrix of partial derivatives for h at the point (2,3).

   (b) Assuming that h(2, 3) = 4, use the result of part (a) to find the equation of the tangent plane to the graph of h at the point (2,3). Write your answer in the form z = Ax + By + C.

8. Let f (x, y) = (x + y)² and let R be the region in the plane bounded by the lines x + y = 0, x + y = 1, 2x - y = 0, 2x - y = 1. Use the change of variables u = x + y, v = 2x - y to evaluate the double integral of f over R.

9. Let M be the level surface g(x, y, z) = 3 where g(x, y, z) = x² + y²/4 + z²/9.

   (a) Find the equation of the tangent plane to the surface M at the point (1,2,3).

   (b) Compute the line integral of the vector field F(x, y, z) = (2x, y/2, 2z/9) over the curve C, where C is parametrized by x(t) = (cos⁴t², sin⁴t², t + 1), 0 < t < (2$\pi$)^1/2.