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Mth 254 Exam - Winter 2006 - Prof. B. Peterson, Exams of Calculus

Oregon State University (OSU)Calculus

Prof. B. Peterson

A past exam for a mathematics course, specifically mth 254, which was held in winter 2006. The exam consists of multiple-choice problems related to various mathematical concepts such as critical points, calculus, and curves. Students were allowed to use a note sheet, a simple scientific calculator, and were not allowed to use books or other notes during the exam. The exam contained a total of 16 problems, with 8 problems worth 8 points each and 8 problems worth 14 points each.

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Mth 254 Exam Winter

2006 Name:

Bent Petersen 254w2006-exam.tex

Date: 9:30 AM Wed, March 22 2006. Location: Kidder 364. Time: 110 min.

•If a scantron is provided with this test then fill in your ID information on the scantron now.

Also enter your name on this test in the space provided above. Do not fold, staple or tear,

etc., the scantron. Return both the entire test and the scantron (separately). If a scantron is

not provided with this test, then ignore the scantron instructions, here and below.

•This test consists of multiple-choice problems. Fill in the answers to the multiple-choice

problems in the boxes below and on the scantron (if one is provided). Depending on your

solution method your answer may appear in a different form from the ones provided on the

test. You are expected to be able to provide the appropriate manipulations to identify the

correct answer.

•You may use one 8.5×11 inch note sheet prepared in advance. You may write on both sides

of your note sheet. Note sheets may not be shared. If you do not bring a note sheet you will

have to do without any help notes. You may not use any books, notebooks, additional note

sheets nor note cards.

•You may use a simple scientific calculator or a modest graphics calculator on this test and

you are expected to have one available. An overly elaborate calculator, laptop, handheld or

notebook computer, or any device capable of extensive symbolic manipulation (other than

your own brain) will not be allowed. Calculators and other equipment may not be shared.

•During the test be sure to check the board occasionally for corrections.

•There are 8 multiple-choice problems worth 8 points each and 8 multiple-choice problems

worth 14 points each (176 points).

Multiple-choice problems: 8 problems, 8 points each.

Problem 1. How many critical poin ts does the function

f(x, y) = 8xy −2y2−x4

have?

A.) 1B.) 2

C.) 3D.) 4E.) None of the foregoing.

←Mark answer here and on the scantron (Problem 1).

Discover Exams of Calculus Oregon State University (OSU)

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Download Mth 254 Exam - Winter 2006 - Prof. B. Peterson and more Exams Calculus in PDF only on Docsity!

Mth 254 Exam

Winter 2006 Name: Bent Petersen 254w2006-exam.tex Date: 9:30 AM Wed, March 22 2006. Location: Kidder 364. Time: 110 min.

If a scantron is provided with this test then fill in your ID information on the scantron now. Also enter your name on this test in the space provided above. Do not fold, staple or tear, etc., the scantron. Return both the entire test and the scantron (separately). If a scantron is not provided with this test, then ignore the scantron instructions, here and below.
This test consists of multiple-choice problems. Fill in the answers to the multiple-choice problems in the boxes below and on the scantron (if one is provided). Depending on your solution method your answer may appear in a different form from the ones provided on the test. You are expected to be able to provide the appropriate manipulations to identify the correct answer.
You may use one 8. 5 × 11 inch note sheet prepared in advance. You may write on both sides of your note sheet. Note sheets may not be shared. If you do not bring a note sheet you will have to do without any help notes. You may not use any books, notebooks, additional note sheets nor note cards.
You may use a simple scientific calculator or a modest graphics calculator on this test and you are expected to have one available. An overly elaborate calculator, laptop, handheld or notebook computer, or any device capable of extensive symbolic manipulation (other than your own brain) will not be allowed. Calculators and other equipment may not be shared.
During the test be sure to check the board occasionally for corrections.
There are 8 multiple-choice problems worth 8 points each and 8 multiple-choice problems worth 14 points each (176 points).

Multiple-choice problems: 8 problems, 8 points each.

Problem 1. How many critical poin ts does the function

f (x, y) = 8xy − 2 y^2 − x^4

have?

A.) 1 B.) 2

C.) 3 D.) 4 E.) None of the foregoing.

Problem 2. The point (2, 4) is a critical point of the function

f (x, y) = 8xy − 2 y^2 − x^4.

Use the second derivative test to classify the critical point (2, 4).

A.) local max B.) local min

C.) saddle D.) global min E.) None of the foregoing.

← Mark answer here and on the scantron (Problem 2).

Problem 3. A moving particle has position vector given by

~r(t) = 5 cos(t)~i + 5 sin(t)~j − 12 t ~k.

Find the speed of the particle.

A.) 12 B.) 22

C.) 13 D.)

194 E.) None of the foregoing.

← Mark answer here and on the scantron (Problem 3).

Problem 4. Find the slope of the tangent to the curve

x^5 y + 2 x^4 y^3 − x^2 y^5 = 2

at the point (1, 1).

A.) − 12 B.) − 5

C.) 6 12 D.) − 112 E.) None of the foregoing.

← Mark answer here and on the scantron (Problem 4).

Problem 5. Consider a particle moving with velocity ~v and acceleration ~a and let v = ‖~v‖

be the speed. If the speed is constant then

A.) ~v and ~a are parallel B.) ~v and ~a are perpendicular

C.) ~v is constant D.) ~a = ~ 0

E.) None of the foregoing.

Multiple-choice problems: 8 problems, 14 points each.

Problem 9. If a > 0 find the maximum curvature of the graph of eax, −∞ < x < ∞.

A.)^3

√ 3

2 a B.)^

2 a 3 √ 3

C.)^3 a

√ 3

2 D.)^

2

3 a√ 3 E.)^ None of the foregoing.

← Mark answer here and on the scantron (Problem 9).

Problem 10. Find the maximum of f (x, y) = 3x + 2y subject to the constraint

x^2 4

y^2 8

A.) 2

17 B.)

C.) 4 D.) 8 E.) None of the foregoing.

← Mark answer here and on the scantron (Problem 10).

Problem 11. Find the surface area of that part of the cone z =

x^2 + y^2 which lies above the disk bounded by the circle x^2 + y^2 − 2 y = 0.

A.) π B.)

2 π

C.) 2 π D.) 2

2 π E.) None of the foregoing.

← Mark answer here and on the scantron (Problem 11).

Problem 12. Evaluate the iterated integral

0

y

ey/x^ dx dy

by first changing the order of integration.

A.) e B.) e + 1

C.) e − 1 D.) (e − 1)/ 2 E.) None of the foregoing.

Problem 13. The twisted cubic ~r(t) = 〈t, t^2 , t^3 〉 intersects the plane 4 x + 2y + z = 24 when

t = 2. If θ is the angle between the tangent to the curve and the normal to the plane at the point of intersection then find cos(θ).

A.) 296 B.)^15

C.) √ 8416 D.) √ 338112 E.) None of the foregoing.

← Mark answer here and on the scantron (Problem 13).

Problem 14. Consider the curve

~r(t) =

et^ cos(t), −et^ sin(t), et

Compute the curvature when t = 0.

A.) 2

3 B.)

C.)

√ 2

√ 3 D.)

√ 2

3 E.)^ None of the foregoing.

← Mark answer here and on the scantron (Problem 14).

Problem 15. Let

f (x, y) = 2x^2 + y^2 − xy − 7 y.

The function f has one critical point. Find the critical point and use the second derivative test to classify it.

Mth 254 Exam - Winter 2006 - Prof. B. Peterson, Exams of Calculus

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Download Mth 254 Exam - Winter 2006 - Prof. B. Peterson and more Exams Calculus in PDF only on Docsity!

Mth 254 Exam

Multiple-choice problems: 8 problems, 8 points each.

Problem 1. How many critical poin ts does the function

A.) 1 B.) 2

C.) 3 D.) 4 E.) None of the foregoing.

Problem 2. The point (2, 4) is a critical point of the function

A.) local max B.) local min

C.) saddle D.) global min E.) None of the foregoing.

Problem 3. A moving particle has position vector given by

A.) 12 B.) 22

C.) 13 D.)

194 E.) None of the foregoing.

Problem 4. Find the slope of the tangent to the curve

A.) − 12 B.) − 5

C.) 6 12 D.) − 112 E.) None of the foregoing.

Problem 5. Consider a particle moving with velocity ~v and acceleration ~a and let v = ‖~v‖

A.) ~v and ~a are parallel B.) ~v and ~a are perpendicular

C.) ~v is constant D.) ~a = ~ 0

E.) None of the foregoing.

Multiple-choice problems: 8 problems, 14 points each.

Problem 9. If a > 0 find the maximum curvature of the graph of eax, −∞ < x < ∞.

A.)^3

2 a B.)^

C.)^3 a

2 D.)^

3 a√ 3 E.)^ None of the foregoing.

Problem 10. Find the maximum of f (x, y) = 3x + 2y subject to the constraint

A.) 2

17 B.)

C.) 4 D.) 8 E.) None of the foregoing.

Problem 11. Find the surface area of that part of the cone z =

A.) π B.)

C.) 2 π D.) 2

2 π E.) None of the foregoing.

Problem 12. Evaluate the iterated integral

A.) e B.) e + 1

C.) e − 1 D.) (e − 1)/ 2 E.) None of the foregoing.

Problem 13. The twisted cubic ~r(t) = 〈t, t^2 , t^3 〉 intersects the plane 4 x + 2y + z = 24 when

A.) 296 B.)^15

C.) √ 8416 D.) √ 338112 E.) None of the foregoing.

Problem 14. Consider the curve

A.) 2

3 B.)

C.)

√ 3 D.)

3 E.)^ None of the foregoing.

Problem 15. Let

A.) (1, 4) local maximum B.) (1, 4) local minimum

C.) (1, 4) saddle point D.) (0, 0) local minimum E.) None of the foregoing.

Problem 16. For what values of a does

A.) all a B.) | a | > 2

C.) | a | < 2 D.) no a E.) None of the foregoing.