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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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