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**preview**shows pages 1-2. to view the full**7 pages of the document.**9/30/2014 FIRST HOURLY Math 21a, Fall 2014

Name:

MWF 9 Oliver Knill

MWF 9 Chao Li

MWF 10 Gijs Heuts

MWF 10 Yu-Wen Hsu

MWF 10 Yong-Suk Moon

MWF 11 Rosalie Belanger-Rioux

MWF 11 Gijs Heuts

MWF 11 Siu-Cheong Lau

MWF 12 Erick Knight

MWF 12 Kate Penner

TTH 10 Peter Smillie

TTH 10 Jeﬀ Kuan

TTH 10 Yi Xie

TTH 11:30 Jeﬀ Kuan

TTH 11:30 Jameel Al-Aidroos

•Start by printing your name in the above box

and please check your section in the box to

the left.

•Do not detach pages from this exam packet

or unstaple the packet.

•Please write neatly. Answers which are illeg-

ible for the grader cannot be given credit.

•Show your work. Except for problems 1-3

or problem 9, we need to see details of your

computation.

•All functions can be diﬀerentiated arbitrarily

often unless otherwise speciﬁed.

•No notes, books, calculators, computers, or

other electronic aids can be allowed.

•You have 90 minutes to complete your work.

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Total: 100

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Problem 1) (20 points) No justiﬁcations are needed.

1) T F The surface −x2+y2+z2=−1 is a one-sheeted hyperboloid.

2) T F The equation y= 3x+ 2 in space deﬁnes a plane.

3) T F Whenever |~r ′(t)|= 1 then |~

T′(t)|= 1.

4) T F The length of the vector projection Proj~v(~w) is smaller than or equal to the

length of ~w.

5) T F The velocity vector of ~r(t) = ht, t, tiat time t= 2 is the same as the velocity

vector at t= 1.

6) T F If ~v ×~w =~w ×~v then ~v, ~w are parallel.

7) T F The vector h−2,1,0iis perpendicular to the line h1 + t, 2t, 3ti.

8) T F The point given in spherical coordinates as ρ= 3, φ = 0, θ =πis the same

point as the point ρ= 3, φ = 0, θ = 0.

9) T F The parametrized curve ~r(t) = h0,3 cos(t),5 sin(t)iis an ellipse.

10) T F The curvature of the line ~r(t) = ht, t, tiis √3 everywhere.

11) T F If |~v ×~w|=~v ·~w then either ~v is parallel to ~w or perpendicular to ~w.

12) T F If the dot product between two unit vectors ~v, ~w is −1, then ~v =−~w.

13) T F Writing ~

k=h0,0,1i, we have |(~

k×~v)×~w| ≤ |~v||~w|for all vectors ~v, ~w.

14) T F The curvature of a curve ~r(t) is given by κ(t) = |~

T′(t)|/|~r ′(t)|. If |~

r′(t)|= 1

for all times, then κ(t) = |~r ′′(t)|.

15) T F The arc length of the curve hsin(t/2),0,cos(t/2)ifrom t= 0 to t= 2πis

equal to 2π.

16) T F If L, K are skew lines in space, there is a unique plane which is equidistant

from L, K.

17) T F The curve ~r(t) = ht, t2,1−tiis the intersection curve of a plane x+z= 1

and y=x2

18) T F The lines ~r1(t) = h5 + t, 3−t, 2−tiand ~r2(t) = h6−t, 2 + t, 1−2tiintersect

at (6,2,1) perpendicularly.

19) T F The vector h3/13,12/13,4/13iis a unit vector.

20) T F ~v ×(~v ×~u) = ~

0 for all vectors ~u, ~v.

Total

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