Solutions to Midterm 1a MA 225 B1 Spring 2011
Question 1
(i) b c is a vector, so we can take its dot product with a. Since the dot product of two vectors is
a scalar, this quantity is a scalar.
(ii) Both (a b) and (c d) are scalars, so it does not make se
Calc I & II review MA 225 B1 Spring 2011
The purpose of these exercises is to emphasize that you are expected to know the material from
Calculus I and II, and to highlight concepts that I feel are particularly important. It is not meant to
be exhaustive,
MA 225 PRACTICE FINAL SOLUTIONS
1. (12 points) Answer the following questions about 3D vector geometry.
(a) (3 pts) Find a vector which is normal to the plane 2x 3z = 1.
Solution: Copying the coefficients of x, y and z and pasting them into a vector gives
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MA 225 PRACTICE FINAL
1. (12 points) Answer the following questions about 3D vector geometry.
(a) (3 pts) Find a vector which is normal to the plane 2x 3z = 1.
(b) (3 pts) Find the angle between the vectors h1, 0, 1i and h1, 1, 0i.
(c) (3 pts) If v and w
FINAL EXAM
Math 21a, Spring 03
Name:
Start by printing your name in the above box and
check your section in the box to the left.
Try to answer each question on the same page as
the question is asked. If needed, use the back or
next empty page for work.
Solutions to Quiz 1 MA 225 B2 Spring 2011
Question 1
(i) 5, 22, 7
(ii) 1
(iii) 1, 2, 7
(iv) They are not orthogonal, because their dot product is nonzero. They are not parallel, because
their cross product is nonzero. Hence: neither.
Question 2 The cross
Solutions to Calc I & II review MA 225 B1 Spring 2011
If you got (m)any of these wrong, dont worry! It has been at least a month since you took Calc II,
and longer since you had Calc I. Just be aware that much of what we will do in this class will rely on
Solutions to Quiz 3 MA 225 B2 and B3 Spring 2011
Question 1 A parameterization of the line is
r(t) = (1 t) 0, 2, 3 + t 1, 6, 4 = t, 2 + 4t, 3 + t ,
Thus,
1
f (r(t)|r (t)|dt =
f (x, y, z )ds =
C
0
1
18
0 t 1.
t2 (2 + 4t)(3 + t)dt = 63 18/10.
0
Question 2 T
Quiz 1 MA 225 B2 Spring 2011
Instructor: Margaret Beck
TF:
Man-Ho Ho
Date:
February 2, 2011
Name:
BU ID:
Score: 1.
(out of 24)
2.
(out of 14)
3.
(out of 42)
4.
(out of 20)
Total:
(out of 100)
Instructions: Please write clearly and show all work. No credit
Quiz 3 MA 225 B2 Spring 2011
Instructor: Margaret Beck
TF:
Man-Ho Ho
Date:
April 27, 2011
Name:
BU ID:
Score: 1.
(out of 20)
2.
(out of 20)
3.
(out of 20)
4.
(out of 20)
5.
(out of 20)
Total:
(out of 100)
Instructions: Please write clearly and show all wo
Solutions to Quiz 2 MA 225 B2 Spring 2011
Question 1 If we test the approach to the origin along a line of the form y = mx, we nd
5mx4
5m
=
,
4 ) x4
(1 + 3m
(1 + 3m4 )
which is dierent for dierent values of m. Hence, the limit does not exist.
Question 2 N
Quiz 2 MA 225 B2 Spring 2011
Instructor: Margaret Beck
TF:
Man-Ho Ho
Date:
March 9, 2011
Name:
BU ID:
Score: 1.
(out of 20)
2.
(out of 20)
3.
(out of 45)
4.
(out of 15)
Total:
(out of 100)
Instructions: Please write clearly and show all work. No credit wi
Midterm 2a MA 225 B1 Spring 2011
Instructor: Margaret Beck
TF:
Man-Ho Ho
Date:
March 31, 2011
Name:
BU ID:
Score: 1.
(out of 10)
2.
(out of 10)
3.
(out of 10)
4.
(out of 10)
5.
(out of 10)
6.
(out of 10)
7.
(out of 10)
8.
(out of 10)
9.
(out of 10)
10.
(o
Solutions to Midterm 2a MA 225 B1 Spring 2011
Question 1 Dierentiating both sides of the equation with respect to y , we obtain
z + yzy = zy /(x + z ).
Solving this equation for zy , we nd
z
z (x + z )
=
.
y
1 y (x + z )
Question 2 The linearization is gi
Midterm 1a MA 225 B1 Spring 2011
Instructor: Margaret Beck
TF:
Man-Ho Ho
Date:
February 15, 2011
Name:
BU ID:
Score: 1.
(out of 10)
2.
(out of 10)
3.
(out of 10)
4.
(out of 10)
5.
(out of 20)
6.
(out of 15)
7.
(out of 20)
8.
(out of 5)
Total:
(out of 100)