MATH 162.02
Solutions for HW #5
Exercise 12.2.36:
(a) Find the unit vectors that are parallel to the tangent line to the parabola
y = 2 sin x at the point (/6, 1).
(b) Find the unit vectors that are perpendicular to the tangent line.
(c) Sketch the curve
Math 21a: Multivariable calculus
Distances overview
DISTANCE POINT-POINT (3D). If P and Q are two points, then
d(P, Q) = |P Q|
is the distance between P and Q. We use the notation |v| instaed
of |v| in this handout.
DISTANCE POINT-PLANE (3D). If P is a po
W%mtm>
.t. _,. . I
7-14 Sketch the curve with the given v
an ath the direction in which I incre
7- Pi!) = (sin I. I)
5- "(f) = (1.2 t.2r)
ector equation. Indicate with
ases.
B. rtt) = (Ari)
10. l'(l) = (sin 17!, t, cos m)
H. rtr) = (Leosr. 25in» 11 rm =
FINAL EXAM
CALCULUS III
MONDAY DECEMBER 16, 2013
o The use of class notes, book, formulae sheet, calculator is not permitted.
0 In order to get full credit, you must show all your work.
0 Please write solution to a problem in the space provided.
0 You hav
FINAL EXAM
CALCULUS III
MONDAY DECEMBER 16, 2013
The use of class notes, book, formulae sheet, calculator is not permitted.
In order to get full credit, you must show all your work.
Please write solution to a problem in the space provided.
You have tw
1. (5 pts.) Find all the fourth roots of the complex number i.
Answer: We have i = e(/2)i . So using the formula from class, the fourth roots are e(/8+k/2)i for
k = 0, 1, 2, 3.
2. Circle the word True or False to indicate your answer. (No explanation need
Calculus III, section 7.
Final Exam 5/10/2012
You can solve as many exercises as you want. The score next to each question reects the
diculty of the question. I recommend that you skim through the test before you choose the
exercises that you want to solv
Calculus III, section 7.
Final Exam 5/13/2010
Solve as many exercises as you can. Exercises marked with are slightly
more dicult and meant for people who are running for an A+; I suggest
you to solve the standard part of the test rst. Moreover, I think it
Calculus III, section 7/8.
2nd Midterm 4/6/2010
Solve as many exercises as you can. Exercises marked with are slightly
more dicult and meant for people who are running for an A+; I suggest
you to solve the standard part of the test rst.
1) Find and sketch
Calculus III, section 7/8.
1st Midterm 02/16/2010
Solve as many exercises as you can. Exercises marked with are slightly
more dicult and meant for people who are running for an A+; I suggest
you to solve the standard part of the test rst.
1) Let P be the
1st Midterm, Calculus 3 Sec 7/8
02/14/2012
You can solve as many exercises as you want. The score next to each question reects the
diculty of the question. I recommend that you skim through the test before you choose the
exercises that you want to solve.
Final Practice 14.1-14.8, Appendix H
Page 1
Final Practice 14.1-14.8, Appendix H
Page 2
Final Practice 14.1-14.8, Appendix H
Page 3
Final Practice 14.1-14.8, Appendix H
Page 4
Final Practice 14.1-14.8, Appendix H
Page 5
Final Practice 14.1-14.8, Appendix
Final Practice 10.3, 10.5, 12.1-12.6, 13.1-13.4, 15.8, 15.9
Page 1
Final Practice 10.3, 10.5, 12.1-12.6, 13.1-13.4, 15.8, 15.9
Page 2
Final Practice 10.3, 10.5, 12.1-12.6, 13.1-13.4, 15.8, 15.9
Page 3
Final Practice 10.3, 10.5, 12.1-12.6, 13.1-13.4, 15.8,
Midterm I Practice Problems 10.3, 10.5, 12.1-12.6, 15.8, 15.9
Page 1
Midterm I Practice Problems 10.3, 10.5, 12.1-12.6, 15.8, 15.9
Page 2
Midterm I Practice Problems 10.3, 10.5, 12.1-12.6, 15.8, 15.9
Page 3
Midterm I Practice Problems 10.3, 10.5, 12.1-12.