Download Spring 1998 Math 320 Final Exam and more Exams Calculus in PDF only on Docsity!
Math 320, Spring 1998 ______________________
Dr. Caldwell name
This two hundred point, two hour final covers much of chapters eleven through fifteen of "Calculus" by Finney
and Thomas. Unless otherwise indicated, each problem is worth eight points. (Suggestion: Scan the whole
test, then relax and start.)
1 Use a calculator to approximate the angle between the curves y = x
and at their rightmost point of
intersection.
2 Use a calculator to approximate the angle between the following two planes (round your radian answer to
two decimal places).
2x 3y + 4x = 5, x + 2y + z = 4
3 Solve the differential equation: with the initial conditions and when t = 0.
4 By considering different paths of approach, show that has no limit as ( x,y ) approaches (0,0).
Final
5 Let f(x,y,z) = e
3 x +4 y
cos 5 z. Find the appropriate second order partials and the determine if f satisfies the
Laplace equation: = 0. Place your results below.
6 On the right is a drawing of several level curves of a
function f(x,y) as well as three points. Draw f as a vector
with initial point at each of these points and clearly indicate
their direction and relative lengths.
7 Let f(x,y) = x
= 25 and P o
a. Find f (at the point P o
). (2 points)
b. Find the equation for the steepest tangent line to this surface at P o
. (3 points)
c. Find the equation for the tangent plane at P o
. (3 points)
12 (a) Find the volume of the solid bounded below by the surface z = , above by the plane z =10, and on the
sides by the cylinder x
= 10,000. (Suggestion: Use cylindrical coordinates.)
(10 points)
(b) Now find the centroid of the above solid.
x = _______ (2 points), y = _______ (2 points), z = _______ (10 points).
13 Evaluate the integral
- Evaluate the transformation:
u = , v = , w =
and integrating over the appropriate region in uvw -space. (Hint: Solve for x, y , and z. )
14 Apply Green's Theorem to evaluate where C is the boundary of the
"triangular" region in the first quadrant enclosed by the x-axis, the line x = 1, and the curve
y = x
15 Use the divergence theorem to find the outward flux of the field across the sphere.
16 Use Stoke's theorem to calculate the circulation of the field
around the square bounded by the lines x = + 1 and y = + 1 in the xy-plane, counterclockwise when
viewed from above.
