MATH 21A Midterm: MATH 21A Harvard 21a Fall 17Practice11
9/27/2017 FIRST HOURLY PRACTICE 11 Math 21a, Fall 2017
Name:
MWF 9 Jameel Al-Aidroos
MWF 9 Dennis Tseng
MWF 10 Yu-Wei Fan
MWF 10 Koji Shimizu
MWF 11 Oliver Knill
MWF 11 Chenglong Yu
MWF 12 Stepan Paul
TTH 10 Matt Demers
TTH 10 Jun-Hou Fung
TTH 10 Peter Smillie
TTH 11:30 Aukosh Jagannath
TTH 11:30 Sebastian Vasey
•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.
1 20
2 10
3 10
4 10
5 10
6 10
7 10
8 10
9 10
10 10
Total: 110
Problem 1) TF questions (20 points)
1
Mark for each of the 20 questions the correct letter. No justifications are needed.
1) T F The
(ρ, θ
2) T F If |~v ×~w|= 0 then ~v =~
0or ~w =~
0.
3) T F The surface z2+ 4y2=x2+ 1 is a two sheeted hyperboloid.
4) T F The surface 4x2−4x+y2−2y−120 = −z2is an ellipsoid.
5) T F The parametrized lines ~u(t) = h1 + 2t, 2−5t, 1 + tiand ~v(t) = h3−4t, −3 +
10t, 2−2tiare the same line.
6) T F The surface sin(x) = zcontains lines which are parallel to the y-axis.
7) T F If ~u ·~v = 0, ~v ·~w = 0 and ~v is not the zero vector, then ~u ·~w = 0.
8) T F The curvature of a curve depends upon the speed at which one travels upon
it.
9) T F Two lines in space that do not intersect must be parallel.
10) T F A line in space can intersect an elliptic paraboloid in 4 points.
11) T F If ~u ×~v = 0 and ~u ·~v = 0, then one of the vectors ~u and ~v is zero.
12) T F
If the velocity vector ~r ′(t) and the acceleration vector ~r ′′ (t) of a curve are
parallel at time t= 1, then the curvature κ(t) of the curve is zero at time
t= 1.
13) T F If the speed of a parametrized curve is constant over time, then the curvature
of the curve ~r(t) is zero.
14) T F The length of the vector projection of a vector ~v onto a vector ~w is always
equal to the length of the vector projection of ~w onto ~v.
15) T F
A quadric ax2+by2+cz2= 1 is contained in the interior of a sphere
x2+y2+z2<100, then the constants a, b, c are all positive and the quadric
is an ellipsoid.
16) T F There is a hyperboloid of the form ax2+by2−cz2= 1 which has a trace
which is a parabola.
17) T F The set of points in space which have distance 1 from the line x=y=z
form a cylinder.
18) T F The velocity vector of a parametric curve ~r(t) always has constant length.
19) T F The volume of a parallelepiped spanned by ~u, ~v, ~w is |(~u ×~v)×~w|.
20) T F The equation x2+y2/4 = 1 in space describes an ellipsoid.
Problem 2a) (3 points)
2
Match the equation with their graphs. No justifications are needed.
I II
III IV
Enter I,II,III,IV here Equation
z= sin(5x) cos(2y)
z= cos(y2)
z=e−x2−y2
z=ex
3
Document Summary
Tth 11:30 sebastian vasey: 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. Mark for each of the 20 questions the correct letter. If |~v ~w| = 0 then ~v = ~0 or ~w = ~0. The surface z2 + 4y2 = x2 + 1 is a two sheeted hyperboloid. The surface 4x2 4x + y2 2y 120 = z2 is an ellipsoid. The parametrized lines ~u(t) = h1 + 2t, 2 5t, 1 + ti and ~v(t) = h3 4t, 3 + 10t, 2 2ti are the same line. The surface sin(x) = z contains lines which are parallel to the y-axis.
