Unformatted text preview: FINAL EXAM
MATH 192, FALL 2003 Write your name, section number, and TA’s name on the test booklet.
No calculators. An 8.5 x 11in sheet of paper with information on both sides is allowed.
Each problem is worth 20 points. ylla 1
1. Consider the integral / / mdemdy.
_ 0 y 2 (a) Evaluate the integral.
(b) Sketch the region of integration.
(c) Reverse the order of integration. 2. Set up a triple integral (or a sum of triple integrals) to compute the volume of the region bounded below-
by the plane 2 = 0, above by the sphere 2:2 + y2 + 22 = 4, and on the sides by the cylinder 1:2 + 312‘: 1. Use
cylindrical coordinates and the order of integration drdde. Sketch the region. Do not evaluate the integral. 3. Consider the integral / f / cosz(¢) dV, where D is the region bounded below by the plane 2 = 0,
D above by the cone 45 = I, and on the sides by the sphere p = 2. (Here (15 and p are spherical coordinates.
3 (a) Set up the triple'integral in spherical coordinates using the order of integration dpqudQ.
(b) Evaluate the integrals. °° ln(n) (:2: — 5)" 4. Find all values of m for which the power series 2 n 32” +1 converges.
tan—1(mz) — sin(m2)
, 8. -Evaluate lim —-————-—
5 ( ) M an
(b) Does the series “21 m converge?
6. The following two lines are given: L1 : :1: = 1 — 2t, '9 = 4t, 2 = —8 —— 3t and L2 : .1: = 4 + s,
y=—1—3s, z= ——6+23.
(a) Do the lines intersect? If yes, ﬁnd the intersection point.
(b) Find the plane that contains both lines.
7. The acceleration vector of a particle in three dimensional space is a(t) = —i —j — 6tk. The velocity of the particle at time t = 0 is v(0) = 0. The particle is at the point (10, 2, 5) at time t = 0. Calculate the
position vector r(t) of the particle as a function of t. Where (at what point) is the particle at time t = 2? 8. Consider the function f(a:, y) = 22y — 2:2 — 23,;2 + 3:: + 4.
(a) Find all critical points. Find the local maxima and local minima, and all points where they occur.
(b) Find the tangent plane to the graph surface 2 = f (2:,y) at the point (1, 0, 6). 9. Consider the function f (9:, y) = my2 + ycos (:r — 1).
(a) Find the linearization at the point (1,2). .
(b) Find an upper bound for the magnitude |E| of the error in the approximation of f by the linearization
over the rectangle |z — 1| 5 0.1, .Iy — 2| 5 0.1. 10. A plane cuts a sphere of radius R. The plane is at a distance h. from the center of the sphere. (We
assume that R > h > 0.) Set up and evaluate a triple integral computing the volume of the region that
contains the center of the sphere, and is bounded by the plane and by the sphere. ...
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- Fall '06