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Math 254Exam 1
April 28, 2015
Name:________________Recitation: ______
Form 1
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xkcd.com
Math 254 Exam 1 - Form 1 Apr 28, 2015
Please read the directions carefully. Good luck!
Written Portion Unjustified answers will receive no credit.
(10pts) Problem 1:A box weighing 100N sits on a ramp that makes an angle of θ = 5π
6radians with the ground.
Decompose the gravity force vector (~F ) into components parallel (~F‖) to and perpendicular
(~F⊥) to the ramp.
θ
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Math 254 Exam 1 - Form 1 Apr 28, 2015
(10pts) Problem 2:A golfer hits a golf ball due North with an initial speed of 88 m/s, at an angle of π
16radians
above the horizontal. A wind blowing due West imparts an acceleration of 0.67 m/s2. Give theacceleration, velocity, and position vectors for the trajectory of the ball. You may use −9.8m/s2
for gravity, and otherwise round to two decimals places throughout. Please be explicit about theorientation you choose for the problem.
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Math 254 Exam 1 - Form 1 Apr 28, 2015
(10pts) Problem 3:Find the curvature function κ(t) for the given vector function. You must actually calculate thecurvature and show all your work to receive credit.
~r(t) = 〈16 cos(t) + 3, 16 sin(t)− 4, 3〉
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Math 254 Exam 1 - Form 1 Apr 28, 2015
Multiple Choice, 3pts ea.
1. Which equation is a vector equation for the tangent line to ~r(t) = 〈1− t2, 5t, 2t3〉 at the point(−3, 10, 16)?
A. ~L(t) = 〈−3 + 4t, 10 + 5t, 16 + 24t〉
B. ~L(t) = 〈−3− 4t, 10 + 5t, 16 + 24t〉
C. ~L(t) = 〈−4− 3t, 5 + 10t, 24 + 16t〉D. ~L(t) = 〈−4 + 3t, 5 + 10t, 24 + 16t〉E. None of these is correct.
2. Find the magnitude of the torque on the pictured knee if θ = π/6.
A. 39.2 Nm
B. 19.6 Nm
C. 19.6̂
D. 33.95 Nm
E. None of these is correct.
3. Suppose the position of a particle at time t is given by ~r(t) = 〈−5 cos(t), 3 sin(t), 4 sin(t)〉. Whatis the speed of the particle?
A. 0
B. 5
C. -5
D. 25
E. None of these is correct.
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Math 254 Exam 1 - Form 1 Apr 28, 2015
For #s 4 – 9, let ~u = 〈6, 2,−3〉 , ~p = 〈4, 9,−2〉 , ~w = 〈2,−4, 0〉.
4. Determine whether ~u, ~p are orthogonal, parallel, or neither.
A. ~u, ~p are orthogonal.
B. ~u, ~p are parallel.
C. ~u, ~p are neither orthogonal nor par-allel.
D. There is not enough information.
5. Calculate 3~u− 2~p+ 12~w.
A. 〈11,−14,−5〉B. 〈8, 18,−4〉C. 〈0, 0, 0〉
D. 〈−11, 14, 5〉E. None of these is correct.
6. Give a unit vector in the direction of ~p.
A. 1√101〈4, 9,−2〉
B. 〈4, 9,−2〉C.⟨25, 910, −1
5
⟩
D.⟨√
32, 0,−1
2
⟩
E. None of these is correct.
7. Find the angle θ between ~u, ~w.
A. θ ≈ 1.443 radians
B. θ = π2
radians
C. θ ≈ 1.051 radians
D. There is not enough information.
E. None of these is correct.
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Math 254 Exam 1 - Form 1 Apr 28, 2015
8. Find the orthogonal projection of ~p onto ~w: proj~w~p.
A. −75〈2,−4, 0〉
B. 75〈2,−4, 0〉
C. 75〈4, 9,−2〉
D. −√32〈2,−4, 0〉
E. None of these is correct.
9. Find the cross product, ~u× ~p.
A. 〈23,−24, 46〉B. ~u, ~p are parallel.
C. 〈23, 24, 46〉
D. 〈−23, 24,−46〉E. None of these is correct.
10. Which integral represents the arc-length of the polar curve r = sin(2θ) for π2≤ θ ≤ π?
A.
∫ π
π2
√5 dθ
B.
∫ π
π2
√sin2(2θ) + 4 cos2(2θ) dθ
C.
∫ π
π2
√sin(2θ) + 2 cos(2θ) dθ
D.
∫ π
π2
√sin2(2θ) + 1 dθ
E. None of these is correct.
11. Find the work done by a force ~F = 5ı̂+ 12̂− 13k̂ (Newtons) to move a particle from P (1, 2, 0)to Q(6, 2,−4) (meters).
A. 0 Joules
B. 77 Joules
C. 52 Joules
D. 36 Joules
E. None of these is correct.
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Math 254 Exam 1 - Form 1 Apr 28, 2015
12. Which graph would tell you the curvature at each value of x for the graph of f below?
2 4 6
f
A. −6 −4 −2 2
B.2 4 6
C.−4 −2 2 4
D.
2 4
13. I certify that I have correctly filled out all the information on the front of the scantron, includingthe Form Number.
A. Yes.
B. No. Please take 1 point off my total exam score.
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Potentially Helpful Information
• The solutions to ax2 + bx+ c = 0 are x =−b±
√b2 − 4ac
2a
• Torque & Work: ~τ = ~r × ~F , W = ~F · ~d
• Curvature: If ~T is the unit tangent vector, curvature is κ(t) = 1||~r ′(t)||
∣∣∣∣∣∣d~Tdt
∣∣∣∣∣∣
Alternatively, if ~v is your velocity vector and ~a is your acceleration, then the curvatureof your path is κ = ||~a×~v||
||~v||3
• Unit Circle:
(1,0),0
(√3
2 , 12
), π
6
(√2
2 ,√
22
), π
4
(12 ,
√3
2
), π
3
(0,1),π2
• Trig Identities: sin2 θ + cos2 θ = 1, 1 + tan2 θ = sec2 θ
Double-Angle: sin 2θ = 2 sin θ cos θ, cos 2θ = cos2 θ − sin2 θsin2 θ = 1
2(1− cos 2θ), cos2 θ = 1
2(cos 2θ + 1)
• Area:
b
Paraellelogram
h
A = bh
b
h
A = 12bh
b1
b2
A = h(b1+b2
2
)
hr
A = πr2
• Volume & Surface Area:
V = 43πr3
SA = 4πr2V = πr2h
SA = πr2 + 2πrh
V = 13πr2h
SA = πr2 + πr√r2 + h2
• 1 + 2 = 3
• e0 = 1, ln(1) = 0
• You are intelligent, capable, and good at math!
