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**preview**shows pages 1-2. to view the full**6 pages of the document.**10/2/2013 FIRST HOURLY PRACTICE IV Math 21a, Fall 2013

Name:

MWF 9 Oliver Knill

MWF 9 Chao Li

MWF 10 Gijs Heuts

MWF 10 Adrian Zahariuc

MWF 10 Yihang Zhu

MWF 11 Peter Garﬁeld

MWF 11 Matthew Woolf

MWF 12 Charmaine Sia

MWF 12 Steve Wang

MWF 14 Mike Hopkins

TTH 10 Oliver Knill

TTH 10 Francesco Cavazzani

TTH 11:30 Kate Penner

TTH 11:30 Francesco Cavazzani

•Start by printing your name in the above box

and 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,

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 time to complete your

work.

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Problem 1) True/False (TF) questions (20 points)

Mark for each of the 20 questions the correct letter. No justiﬁcations are needed.

1) T F The surface x2+y2+z2+ 2z= 0 is a sphere.

2) T F The length of the vector h1,2,2iis an integer.

3) T F The vector h3,4iappears as a velocity vector of the curve ~r(t) =

hcos(5t),sin(5t)i. Namely, there is a tsuch that ~r ′(t) = h3,4i.

4) T F If ~

Tis the unit tangent vector, ~

Nis the unit normal vector, and ~

Bis the

binormal vector, then ~

B×~

N=~

T.

5) T F The curvature of a larger circle r= 2 is greater than the curvature of a

smaller circle r= 1/2.

6) T F The surface x2−y2−z2−1 = 0 is a one sheeted hyperboloid.

7) T F The function f(x, y) = y2−x2has a graph that is an elliptic paraboloid.

8) T F Let ~r(t) be a parametrization of a curve. If ~r(t) is always parallel to the

tangent vector ~r ′(t), then the curve is part of a line through the origin.

9) T F If Proj~

k(~u) is perpendicular to ~u, then ~u is the zero vector.

10) T F If Proj~

k(~u) is perpendicular to ~u, then Proj~

k(~u) is the zero vector.

11) T F If ~u ×~v =~

0 then ~u =~

0 or ~v =~

0.

12) T F There are two vectors ~a and ~

bsuch that the scalar projection of ~a onto ~

bis

100 times the magnitude of ~

b.

13) T F The curve ~r(t) = hcos(t), et+ 10, t2i,2≤t≤6 and the curve ~r(t) =

hcos(2t), e2t,4t2i,1≤t≤3 have the same length.

14) T F The equation ρsin(φ)−2 sin(θ) = 0 in spherical coordinates deﬁnes a two

sheeted hyperboloid.

15) T F If triple scalar product of three vectors ~u, ~v, ~w is larger than |~u ×~v|then

|~w|>1.

16) T F The distance between the x-axis and the line x=y= 1 is √2.

17) T F The vector h−1,2,3iis perpendicular to the plane x−2y−3z= 9.

18) T F The curve ~r(t) = t3h1,2,3iis a line.

19) T F The point (1,1,−√3) is in spherical coordinates given by (ρ, θ, φ) =

(√5, π/4,2π/3).

20) T F If the cross product satisﬁes (~v ×~w)×~v =~

0 then ~v and ~w are orthogonal.

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