Each Problem Is Worth 10 Points. the Problems Are Listed in Random Order

NAME:______

MATH 224

FINAL EXAM

Don Lawrence

May 7, 2003

No notes or calculators allowed. You do not have to simplify your answers (that is, you may stop after you’ve done all the 224 calculations.)

Each problem is worth 10 points. The problems are listed in random order.

Here are the formulas I said I would give you.

In spherical coordinates, , , and .

Trig identities: and .

(1)

(2) S is the surface of the tetrahedron with vertices (0,0,0), (1,0,0), (0,2,0), and (0,0,4), oriented outward. Find , where .

(3) If , find .

(4) Find , where , and C is this semi-circle.

(5) Find maxes and mins of on the circle . Then discuss absolute maxes and mins of on the circle.

(6) Suppose that , , , and . Find a formula for .

(7) Find , where and C is this circle.

(8) Make a contour diagram for , with contours labeled z=-1, z=0, and z=1. Then sketch the graph of , and write a phrase describing the shape of the surface you attempted to draw.

(9) Assign one of these words to each quantity: up, down, positive, negative, zero. No explanation is required.

(a)

(b)

(c)

(d)

(10) Find , where and C is this circle.

(11) (a) Find the derivative of at the point in the direction of .

(b) What is the largest directional derivative of at ?

(12) Find , where and S is the result of rotating the curve (for ) around the z-axis, oriented up. [Hint: First parametrize S, using and as your parameters. Notice that corresponding to a fixed value of x, there is a circle on S at height and with radius x. If you can’t find the parametrization, still write down the formula you would have used to compute the flux integral.]

(13) Let V be the set of differentiable real-valued functions of one variable. (That is, V is the set of f(x) that have derivatives.) Then V is a vector space over (you don’t need to verify this.) Define by , for any . Show that T is a linear transformation. (You may use standard properties of derivatives without proving them.)

(14)

(15) W is a solid doughnut with two holes. The three curves divide W’s boundary into two surfaces, and , both oriented outward. Suppose that , , and . Find , , and . Show work, and cite each theorem you use.

(16) Find the critical points of , and use the second derivative test to classify them. Then discuss absolute maxes and mins of .

(17) Describe the motion given by . That is, give an equation for the xy template curve on which the action takes place, then describe how a particle moves along that curve.

(18) Let S be the surface defined by .

(a) Find an equation of the plane tangent to S at (1,-2,-1).

(b) Approximately how high is S over the point (1.1, -2.1)?

(19) Check that does not intersect . Then find the distance between them.

(20) Find the angle between the curve and the plane at their point of intersection.