Docsity
Docsity

Prepare for your exams
Prepare for your exams

Study with the several resources on Docsity


Earn points to download
Earn points to download

Earn points by helping other students or get them with a premium plan


Guidelines and tips
Guidelines and tips

Practice Exam provided, Exams of Calculus

Provided by the professor in order to help study.

Typology: Exams

2024/2025

Uploaded on 05/13/2025

jacob-weber-2
jacob-weber-2 🇺🇸

1 document

1 / 7

Toggle sidebar

This page cannot be seen from the preview

Don't miss anything!

bg1
May 5, 2003 MATH 208 Final Exam Spring Semester, 2003
Name: Section: Instructor:
Instructions:
(1) You must show supporting work to receive credits.
(2) Use exact values whenever possible, e.g. π/4 instead of 0.785398...
(3) Calculators are allowed.
(4) Make sure you have all the 7 pages of questions.
Problem 1 2 3 4 5 6 7 8 9 10 11 12 13 Total
Point 14 16 16 16 16 18 14 18 12 16 14 14 16 200
Credit
1. (14 points) Let f(x, y) = 4x+ 5y.
(a) (10 points) Find the quadratic approximation to f(x, y) at the point (4,3).
(b) (4 points) Use the quadratic approximation of (a) to approximate f(4.1,2.9).
(Continue on Next Page ... )
pf3
pf4
pf5

Partial preview of the text

Download Practice Exam provided and more Exams Calculus in PDF only on Docsity!

May 5, 2003 MATH 208 Final Exam Spring Semester, 2003

Name: Section: Instructor:

Instructions:

(1) You must show supporting work to receive credits.

(2) Use exact values whenever possible, e.g. π/4 instead of 0. 785398 ...

(3) Calculators are allowed.

(4) Make sure you have all the 7 pages of questions.

Problem 1 2 3 4 5 6 7 8 9 10 11 12 13 Total

Point 14 16 16 16 16 18 14 18 12 16 14 14 16 200

Credit

  1. (14 points) Let f (x, y) =

4 x + 5y.

(a) (10 points) Find the quadratic approximation to f (x, y) at the point (4, −3).

(b) (4 points) Use the quadratic approximation of (a) to approximate f (4. 1 , − 2 .9).

  1. (16 points) Let S be the surface given by

6 − z

2

x

2

  • y

2

(a) (4 points) Verify that point P = (1, 1 , 2) is on the surface.

(b) (6 points) Find a normal vector of S at the point P.

(c) (6 points) Write down an equation of the tangent plane to S at P.

  1. (16 points) Use polar coordinates to evaluate the following integral:

2

0

4 −y

2

0

e

x

2 +y

2

dx dy.

  1. (18 points) Let f (x, y, z) = 2x

2 ln(y) + xe

2 z , let P = (− 2 , 1 , 0) and Q = (2, 5 , −2).

(a) (6 points) Find and simplify 5 f (− 2 , 1 , 0).

(b) (6 points) Find and simplify the rate of change of f at P in the direction from P to Q.

(c) (6 points) Find and simplify the unit vector in the direction in which f decreases most rapidly at P.

  1. (14 points) Consider

1

0

1 −x

2

0

1 −x

2 −y

2

0

(x + z)dzdydx. Change the integral to an iterated triple

integral in the spherical coordinates. (DO NOT EVALUATE the triple integral.)

  1. (18 points) Let

F (x, y) = (x

2

  • 2xy)

i + (x

2

  • 5y + 1)

j.

(a) (6 points) Use a derivative test to verify that the vector field

F (x, y) is a conservative vector field.

(b) (8 points) Find a potential function f for

F (x, y).

(c) (4 points) Find the

C

F (x, y) · d~r, where C is a smooth path from (1, 0) to (0, 1).

  1. (12 points) Consider the integral

2

0

8 x

x

4

f (x, y)dydx.

(a) (4 points) Sketch or describe the region of integration.

(b) (8 points) Switch the order of integration.

  1. (14 points) Suppose

F is a smooth vector field defined everywhere such that div

F = 10. Find the flux of

F out of a closed cylinder (with cover and base) of height 5 and radius 2, centered on the z-axis with base

in the xy-plane.

  1. (16 points) Let

F = (y + 2z)

i + 4x~j + yz

k.

(a) (6 points) Find the curl of

F : 5 ×

F.

(b) (10 points) Use your result from part (a) to find the line integral around the circle of radius 1 in the

xy-plane, centered at the origin, oriented counterclockwise when viewed from above.

(... End )