MATH 222 Name___________________________
Calculus and Analytic Geometry III Student#______________________
Summer 2003

In-Class Examination I

Show all work. If you must continue a problem outside of the space provided, leave clear instructions where to find the additional work. Work outside the area provided will be ignored in assigning (partial) credit if instructions to its location are not found in the space provided. Correct answers without sufficient supporting work will be worth at most 1 point for problems or parts of problems with 5 points or less, and at most 2 points for other problems. Point values of problems follow the question number. Vector or numerical calculations done using a calculator must be documented (“using a calculator we obtain. . .”). If a graph is used (e.g. to decide the direction of a principal normal vector) the graph must be shown as part of the work. Provide exact solutions (e.g. 2p not 6.28..., or arccos(1/3)-1 not 0.23).


Vectors are denoted in bold-face (e.g. u, v, i, j, k), scalars in italic (e.g. v, x, k).

1. [9] Let u = (-1,-5,1) and v = (2,1,1). Compute the following:

a) 2u + 3v




b) u · v





c) u ¥ v




2. [9] Find the angle between the planes

2x + y - 2z = 3 and 2x - y + 2z = 5.














3. [8] Find the equation of the plane tangent to the graph of

f(x,y) = 9x2 - 12x + 16y2 at the point (1,1,13).















4. [6] Classify the locus of the equation

x2 + xz + 4z2 = y

as an elliptic paraboloid or a hyperbolic paraboloid.

5. [10] Consider the function f(x,y,z) = x2 - 9y2 + 4z2

a) Describe in detail the level sets. (Hint: what kind of quadric surfaces are they? Does the answer depend on the level?)










b) Find all first partial derivatives of f.














6. [15] Suppose the position of a particle is subject to a constant acceleration of a(t ) = -10k + 5j, has an initial velocity of v0 = 50j + 20i, and has an initial position of r0 = 1000k.

a) Find the velocity vector as a function of time.








b) Find the equation of motion, that is the position (vector) as a function of time.








c) Find the speed as a function of time.





problem 6 (continued from previous page)

d) Find the unit tangent vector to the particle’s path as a function of time.












e) Find the magnitude of the tangential and normal components of the acceleration at the time t=0.









7. [8] Convert the following into cylindrical and rectangular coordinates:

a) (r, f, q ) = (4,,)













b) r = 5cos(q )

















8. [10] Find the following limits or show they do not exist

a)











b)
9. [10] Suppose P and Q are points on the line L in 3-space. Let A be a point not on the line L. Calculate in two ways the area of the triangle APQ to show that the perpendicular distance from A to the line L is

10. [15] Consider the curve 2y = x2 in the plane.

a) Give a vector equation for the curve.









b) Find the unit tangent vector to the curve at (1,).











c) Find the curvature of the curve at (1,).










problem 10 (continued from previous page)

d) Find the principal unit normal vector to the curve at (1,).












e) Give an equation for the osculating circle to the curve at (1,).

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