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I am taking clac 3 final test this Tuesday and I need help with it.

MAT267 – Final Exam – Fall2014

MAT 267 – Calculus for Engineers-III

Instructor : RUEDEMANN

Final Exam 2014 Form A

Student Name :

Student ID :

Class Time :

Honor Statement

By signing below I confirm that I have neither given nor received any unauthorized assistance on

this exam. This includes any use of a graphing calculator beyond those uses specifically authorized

by the School of Mathematical and Statistical Sciences and my instructor. Furthermore, I agree not

to discuss this exam with anyone until the exam testing period is over. In addition, my calculator’s

memory and menus may be checked at any time and cleared by any testing center proctor or

School of Mathematical and Statistical Sciences instructor.

Signature

Instructions:

1. The exam consists of two parts: multiple choice, worth 60%, and free response (show your work), worth

40%. Please read each problem carefully.

2. There are 10 multiple choice questions worth 6 points each. Please fill in the table provided.

3. Provide complete and well-organized answers in the free response section.

4. Answers in the free response section without supporting work will be given zero credit. Partial credit is

granted only if work is shown.

5. No calculators with Qwerty keyboards or ones like the Casio FX-2, TI-89, TI-92, or TI-nspire that do symbolic

algebra may be used.

6. Proctors reserve the right to check calculators.

7. Please request scratch paper from me if you need it.

8. The use of cell phones is prohibited. TURN YOUR CELL PHONE OFF! Do not allow your cell phone to ring

while you are taking the exam. Do not use the calculator on your cell phone. If a proctor sees you using a

cell phone, they will take your exam and you will be reported to the Dean of Students for cheating.

9. PLEASE NOTE: “Any student who accesses a phone or any internet-capable device during an exam for any

reason automatically receives a score of zero on the exam. All such devices must be turned off and put away

and made inaccessible during the exam.”

MAT267 – Final Exam – Fall2014

Read all directions carefully! Be neat, and box all answers. Points will be deducted for not following

directions, sloppiness or lack of relevant work shown.

PLEASE NOTE: “Any student who accesses a phone or any internet-capable device during an exam for any

reason automatically receives a score of zero on the exam. All such devices must be turned off and put away and

made inaccessible during the exam.”

Section I: Multiple Choice. Write your answers in the table provided. If you think the answer is “none

of the above”, write in E. (6 pts each)

1. Find the angle (in degrees) between the vectors and . (Round to two decimal

places.)

𝐀. 81.41° 𝐁. 40.20° 𝐂. 72.02° 𝐃. 43.71° E. None of these

2. Which one of the following are parametric equations for the tangent line to the curve

𝑥 = 1 + sin 𝑡 , 𝑦 = 𝑒 𝑡 , 𝑧 = 𝑡 + 2

at the point (1,1,2)?

A. 𝑥 = 1 + 𝑡,

𝑦 = 1 + 𝑡,

B. 𝑥 = 1 + 3𝑡 2 ,

C. 𝑥 = 1,

𝑦 = 1 + 2𝑡,

𝑦 = 1 + 𝑡,

D. 𝑥 = 1 + 3𝑡,

𝑧 =2+𝑡

𝑧 = −1 + 𝑡

𝑧 =2−𝑡

𝑦 = 1 − 2𝑡,

𝑧 =1+𝑡

E. None of these

3.

Find the projection of 𝐯 = 〈1,2〉 onto the vector 𝒖 =. That is, find proj𝒖 𝒗.

33 44

𝐀. 〈 5 , 5 〉

11 22

𝐁. 〈25 , 25〉

3

4

𝐂. 〈25 , 25〉

33 44

𝐃. 〈25 , 25〉

E. None of these

MAT267 – Final Exam – Fall2014

4. Suppose 𝑧 = 𝑥 2 + 𝑦 3 𝑥 = 𝑒 𝑠𝑡 , and y = ln(2𝑠 + 5𝑡). Find

𝜕𝑧

𝜕𝑠

(Do not simplify.)

1

A. 2𝑥𝑒 𝑠𝑡 𝑠 + 3𝑦 2 2𝑠+5𝑡

1

B. 2𝑥𝑒 𝑠𝑡 𝑡 + 3𝑦 2 2𝑠+5𝑡

2

C. 2𝑥𝑒 𝑠𝑡 𝑡 + 3𝑦 2 2𝑠+5𝑡

5

D. 𝑥 2 𝑒 𝑠𝑡 𝑡 + 𝑦 3 2𝑠+5𝑡

E. None of these

5 Find the directional derivative of the function 𝑓(𝑥, 𝑦) = 𝑥 2 sin 𝑦 at the point (1, 𝜋) in the direction of

the vector 𝒗 = 5𝒊 + 12𝒋.

12

𝐀. − 13

17

5

𝐁. 13

𝐂 . − 13

𝐃. 1

E. None of these

2

6. Use Green’s theorem to evaluate∫𝐶 (𝑥𝑒 𝑥 )𝑑𝑥 + (𝑥 + ln 𝑦)𝑑𝑦, where C is the circle 𝑥 2 + 𝑦 2 = 1 oriented in

positive direction.

𝐀. 0

𝐁. 𝜋

𝐂 . 2π

𝐃. 1

E. None of these

7. Let 𝐹(𝑥, 𝑦, 𝑧) =. Find the divergence of and curl of 𝐹.

A. div (F) = 12+2𝑥 and curl(F)= 0𝒊 + 0𝒋 + 2𝑦𝒌

B. div (F) = 12+2x and curl(F)= 5𝒊 + 2𝑦𝒋 + 7𝒌

C. div (F) = 14+2x and curl(F)= 0𝒊 + 2𝑦𝒋 + 0𝒌

D. div (F) = 14 and curl(F)= 5𝒊 + 2𝑦𝒋 + 7𝒌

E. None of these

8. For the surface with the parametric equations 𝒓(𝑠, 𝑡) = find an equation of the tangent plane

to the surface at (6,8,3)

A. −18(𝑥 − 6) + 324(𝑦 − 8) − 11(𝑧 − 3) = 0

B. −4(𝑥 + 6) + 3(𝑦 + 8) − 24(𝑧 + 3) = 0

C. −4(𝑥 − 6) + 3(𝑦 − 8) − 24(𝑧 − 3) = 0

D. −18(𝑥 + 6) + 324(𝑦 + 8) − 11(𝑧 + 3) = 0

E. None of these

MAT267 – Final Exam – Fall2014

9. Find the velocity vector of a particle with acceleration 𝒂(𝑡) = and initial velocity 𝒗(0) =

1

𝐀.

𝐁.

1

𝐃.

2

1

𝐂.

𝐄 . None of these.

10. Evaluate the double integral ∬𝐷 2𝑦 𝑑𝐴 where D is the triangular region with vertices (0,0), (3,0), (0,3).

𝐀. 9

𝐁. 18

𝐂 . −18

𝐃. −9

E. None of these

PART II – Free Response. You must show all work for credit. Box your final answers.

11. [10 pts] Find the local maximum and minimum values and saddle point(s), if any of the function

𝑓(𝑥, 𝑦) = 𝑥 2 + 3𝑦 2 + 2𝑥𝑦 − 8𝑥 − 12𝑦

MAT267 – Final Exam – Fall2014

1

12. [10 pts] Let 𝑭(𝑥, 𝑦, 𝑧) = (𝑦𝑧)𝒊 + (𝑥𝑧 + 𝑦) 𝒋 + (𝑥𝑦 + 3𝑧 2 )𝒌

a)

Find a potential function for F .

b.)Use your answer to part (a) above to find

F

dr , where C is a curve from (2, 1, 3) to (1, 1, 5)

C

MAT267 – Final Exam – Fall2014

13. [10 pts] Find the surface area of the part of the plane 2𝑥 − 3𝑦 + 𝑧 = 100 that lies inside of the cylinder

𝑥 2 + 𝑦 2 = 16.

MAT267 – Final Exam – Fall2014

14. [10 pts] Find the flux of the vector field 𝐸 = outward through the sphere 𝑥 2 + 𝑦 2 + 𝑧 2 = 1.

FINAL REVIEW

TOPICS FROM TEST 1

TOPICS FROM TEST 2

TOPICS FROM TEST 3

Answers

MAT 267, S2021, Exam 1

Instructor: Tom Taylor

Print Name:kshumarg@asu.edu

Honor Statement:

By signing below I confirm that I have neither given nor received any unauthorized assistance on this

exam. This includes any use of a graphing calculator beyond those uses specifically authorized by the

School of Mathematical and Statistical Sciences and my instructor. Furthermore, I agree not to discuss

this exam with anyone until the exam testing period is over. In addition, my calculator’s memory and

menus may be checked at any time and cleared by any testing center proctor or School of Mathematical

and Statistical Sciences instructor.

Signature

Date

INSTRUCTIONS Please read the following items carefully before proceeding.

You will have the 50 minutes to complete the exam.

Your webcam must be on and your face visible at all times.

There are both multiple choice (56 points–7points each) and free response (44 points) problems.

IN BOTH CASES SHOW ALL WORK CLEARLY AND NEATLY ORGANIZED IN

THE GIVEN SPACE. ANSWERS WITHOUT CLEARLY STATED SUPPORTING

WORK WILL BE GIVEN ZERO CREDIT.

You may turn in your exam either as a photographs of neatly organized exam problems written on

blank paper or as a photograph of work written on a printed copy of this exam, NOT BOTH. No

credit will be given for unorganized work written on scratch paper.

The returned exams may be either image files (.jpeg, .jpg, .png) or pdf files.

Proctors reserve the right to check calculators. Please follow the instructions of the proctor if asked

to demonstrate your calculator.

If you finish early, you may turn in your exam to the instructor and quietly leave.

Please have your ASU student ID ready to present when turning in your exam.

All electronic devices other than calculators, especially cell phones, must be completely turned off

and put away. Any student attempting to access such a device will automatically receive a score of

zero on the exam.

MAT 267, Test 1, Form kshumarg@asu.edu

Page 1

1.(6 pts) The arc length of the curve C parameterized as ~r(t) = h1 + 4t, −5 + 2t, 5 − 5ti, 0 ≤ t ≤ 1 is

R1p

A. 0 (4t + 1)2 + (2t − 5)2 + (5 − 5t)2 dt

√

B. 3 5

√

C. 51

R1

D. 0 h1 + 4t, −5 + 2t, 5 − 5tidt

E. none of the above

MAT 267, Test 1, Form kshumarg@asu.edu

Page 2

p

2. (6 pts) Let S be the cone z = 3 x2 + y 2 . Which of the following is a possible parameterization of S?

A. x = cos(s), y = sin(s), z = 3t2

B. x = t cos(s), y = t sin(s), z = 3t

C. x = 3 cos(s), y = 3 sin(s), z = t

D. x = 3t cos(s), y = 3t sin(s), z = t

E. none of the above

MAT 267, Test 1, Form kshumarg@asu.edu

Page 3

3. (12pts) A rectangular box has dimensions x,y and z in units of feet.If the dimensions of the box must

satisfy the constraint 3x + 4y + 6z = 12, find the dimensions that produce the largest possible volume.

MAT 267, Test 1, Form kshumarg@asu.edu

Page 4

~

4. (6 pts) Find the curl of F(x,

y, z) = hx2 z, x2 y, 2xyzi at the point (2, −1, −1).

A. h−4, 5, 2i

B. h−4, 2, −4i

C. h0, 0, 0i

D. h2xz, x2 − 2yz, 2xyi

E. none of the above

MAT 267, Test 1, Form kshumarg@asu.edu

Page 5

5. (14 pts) Evaluate the surface integral

hs, t, 3s + 2ti, 0 ≤ s ≤ 1, 0 ≤ t ≤ 2.

RR

MAT 267, Test 1, Form kshumarg@asu.edu

S (x

+ yz) dS where S is the parametric surface ~r(s, t) =

Page 6

6. (6 pts) Find the directional derivative of

f (x, y) = 4×2 yz − 2xy + 2xz at the point (−2, 0, −2) and in the direction of ~v = h2, 1, 2i.

√

A. −27 2

B. −108

C. − 44

3

D. −44

E. none of the above

MAT 267, Test 1, Form kshumarg@asu.edu

Page 7

7. (6 pts) If w = 3x + y, x=st2 and y = sin(t) − s, which of the expressions represents

∂w

∂s ?

A. 3(−1) + 1(t2 )

B. 3 + 1

C. h3, 1i

D. 3(t2 ) + 1(−1)

E. none of the above

MAT 267, Test 1, Form kshumarg@asu.edu

Page 8

~

8. (14 pts) Find the flux of the vector field F(x,

y, z) = hz, x, yi through the parametric surface

~r(s, t) = hs, t, 3s + 2ti, 0 ≤ s ≤ 1, 0 ≤ t ≤ 2. Assume the surface has upward orientation.

MAT 267, Test 1, Form kshumarg@asu.edu

Page 9

√

√

2

2

2

2

9. (6 pts) Let E be the region bounded

by the

p two hemispheres y = 4 − x − z and y = 25 − x − z

RRR

2

2

2

and the xz-plane. The triple integral

x + y + z dV in spherical coordiantes is

E

RπRπR5

A. 0 0 2 ρ2 sin(φ)dρdθdφ

π

2

B.

RπR

C.

RπRπR5

D.

RπR

0

0

0

− π2

0

π

2

− π2

R5

2

2

ρ3 sin(φ)dρdθdφ

ρ3 sin(φ)dρdθdφ

R2

4

5ρ4 sin(φ)dρdθdφ

E. none of the above

MAT 267, Test 1, Form kshumarg@asu.edu

Page 10

10. (12 pts) A toy rocket with a mass of 6 kg is launched from the ground at t = 0 and crashes on

the ground (z = 0) some time later. During the flight, its position vector is observed to be ~r(t) =

ht4 + 2t3 , t2 + 2t, 10t − 5t2 i where t is in seconds and distances are in meters.

a)[6 pts] Find the

speed the rocket was traveling when it crashed.(round answer to 2 decimal places)

b)[6 pts] Find

~ that was acting on the rocket when it launched.

the force vector F

MAT 267, Test 1, Form kshumarg@asu.edu

Page 11

~

11 (6 pts) Find the divergence of the vector field F(x,

y, z) = h2xyz, xy 2 , y 2 zi at the point (2, −2, 3).

A. 9

B. −16

C. 2xy + y 2 + 2yz

D. 0

E. none of the above

MAT 267, Test 1, Form kshumarg@asu.edu

Page 12

12 (6 pts) Find the surface area of the parametric surface

~r(s, t) = h−t, 2t, s + 2ti, 0 ≤ s ≤ 2, 0 ≤ t ≤ 4.

√

A. 5

B. h−16, −8, 0i

C. h−2, −1, 0i

√

D. 8 5

E. none of the above

MAT 267, Test 1, Form kshumarg@asu.edu

Page 13

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