MATH 105: PRACTICE PROBLEMS FOR CHAPTER 1
AND CALCULUS REVIEW: SPRING 2010
SOLUTION KEY (PLEASE REPORT ANY ERRORS TO ME)
INSTRUCTOR: STEVEN MILLER (SJM1@WILLIAMS.EDU)
Question 1 : These problems deal with equations of lines.
(1) Find the equation of the l
MATH 105: PRACTICE PROBLEMS FOR CHAPTER 1
AND CALCULUS REVIEW: SPRING 2010
INSTRUCTOR: STEVEN MILLER (SJM1@WILLIAMS.EDU)
Question 1 : These problems deal with equations of lines.
(1) Find the equation of the line going through the points (2,3) and (4,9).
Math 105 - Multivariable Calculus (Miller) -
Practice Midterms - 2010
First Practice Midterm
1. (20 points) Let (, ) = 3 + cos and (, ) = 42 + . Find the derivatives of the following
functions if possible; if it is not possible to nd the derivative state
MATH 105: PRACTICE PROBLEMS FOR CHAPTER 2: SPRING 2010
INSTRUCTOR: STEVEN MILLER (SJM1@WILLIAMS.EDU)
Question 1 : These problems deal with open sets.
(1)
(2)
(3)
(4)
(5)
(6)
Let
Let
Let
Let
Let
Let
= cfw_(, , ) : 32 + 4 2 + 5 2 < 6. Is open?
= cfw_(, ) :
MATH 105: PRACTICE PROBLEMS FOR CHAPTER 5: SPRING 2010
INSTRUCTOR: STEVEN MILLER (SJM1@WILLIAMS.EDU)
Question 1 : Dene the following terms:
(1) What does it mean for a function : 2 to be bounded?
(2) Dene a simple region.
Solution: (1) A function is bound
MATH 105: PRACTICE PROBLEMS FOR CHAPTER 3: SPRING 2010
INSTRUCTOR: STEVEN MILLER (SJM1@WILLIAMS.EDU)
Question 1 : Compute the partial derivatives of order 1 and order 2 for:
(1) (, , ) = + cos() sin().
Solution: We can proceed by brute force, but it helps
MATH 105: PRACTICE PROBLEMS FOR CHAPTER 6 AND
SEQUENCES AND SERIES: SPRING 2010
INSTRUCTOR: STEVEN MILLER (SJM1@WILLIAMS.EDU)
Question 1 : State the change of variable theorem in the plane. How does the element
transform in polar coordinates? How does tr
MATH 105: PRACTICE PROBLEMS FOR CHAPTER 3: SPRING 2010
INSTRUCTOR: STEVEN MILLER (SJM1@WILLIAMS.EDU)
Question 1 : Compute the partial derivatives of order 1 and order 2 for:
(1) (, , ) = + cos() sin().
(2) (, ) = sin( / ).
Question 2 : Compute the second
MATH 105: PRACTICE PROBLEMS AND SOLUTIONS
FOR CHAPTER 2: SPRING 2010
INSTRUCTOR: STEVEN MILLER (SJM1@WILLIAMS.EDU)
Question 1: These problems deal with open sets.
(1) Let = {(, , ) : 32 + 4 2 + 5 2 < 6}. Is open?
Solution: Yes: This is an ellipsoid where
MATH 105: PRACTICE PROBLEMS FOR CHAPTER 6 AND
SEQUENCES AND SERIES: SPRING 2010
INSTRUCTOR: STEVEN MILLER (SJM1@WILLIAMS.EDU)
Question 1 : State the change of variable theorem in the plane. How does the element
transform in polar coordinates? How does tr
MATH 105: PRACTICE PROBLEMS FOR CHAPTER 5: SPRING 2010
INSTRUCTOR: STEVEN MILLER (SJM1@WILLIAMS.EDU)
Question 1 : Dene the following terms:
(1) What does it mean for a function : 2 to be bounded?
(2) Dene a simple region.
Question 2 : Compute
4
2
=
=1
.
Math 381: First Exam
September 26, 2003
1
Read the directions to each problem carefully. Show your work when necessary. This exam has 8 questions on 5 pages and lasts 50 minutes. Name: Pledge:
Problem 1:(6 points) Identify the order of the following parti
Math 381: Final Exam
December 11, 2003 2pm - 5pm
1
Read the directions to each problem carefully. Show your work and answer in complete sentences when appropriate. This exam has 5 questions on 11 pages including this cover, blank pages for work and the ta
Solutions to the Final Examination Math 381: Introduction to Partial Differential Equations Rice University, Fall 2003
Problem 1: a) This is a fourth order equation. b) Many correct responses are possible. An example would be z z + 12 = 3z. x y c) Again,
1
Math 381: Second Exam Due: Monday, November 10, 2003 at the beginning of class. This exam is self-timed. You have a maximum of 3 hours. You may consult your textbook or your class notes, but you may not discuss the content of this exam with anyone but y
1
Math 381: Second Exam
November 5-10, 2003 Solutions Problem 1: We rewrite our problem in polar coordinates because these are adapted to the disk D. Then our boundary condition is g(1, ) = (cos )2 . Following the developments in class, we know that the s
Math 381: First Exam
September 26, 2003 Solutions Problem 1: The equations are second order, fourteenth order and first order, respectively.
1
Problem 2: We compute the derivatives 2y = f (x + at) - g (x - at), x2 2y = a2 f (x + at) - a2 g (x - at). t2 So
Solutions to Homework # 1 Math 381, Rice University, Fall 2003 Hildebrand, Ch. 8, # 1: Part (a). We compute z = f (x + y) + (x - y)f (x + y) x z = -f (x + y) + (x - y)f (x + y). y Subtracting, we eliminate f . . . z z - = 2f (x + y). x y Substitute in fro
Solutions to Homework # 2 Math 381, Rice University, Fall 2003 Problem1: The function is odd, so all of the ak 's are zero, including a0 . We then compute the bk 's to be bk = = 2
sin(kx)dx =
0
-2 -2 (cos(k) - cos(0) = (-1)k - 1 k k
0,
4 k ,
k even . k o
Solutions to Homework # 5 Math 381, Rice University, Fall 2003 Problems 1-4: Since these are polynomials, it is not to hard to compute the answers. First, note that a Legendre polynomial is orthogonal to any polynomial of lower degree, so most of the coef
Solutions to Homework # 7 Math 381, Rice University, Fall 2003 Problem 1: We work each by hand using integration by parts. a) We integrate by parts twice to find that L(eat cos(kt) =
0
e-st eat cos(kt) dt e-(s-a)t cos(kt) dt
=
0
-e-(s-a)t cos(kt) ke-(s-
Solutions to Homework # 6 Math 381, Rice University, Fall 2003 Problem 1: The problem is about Laplace's equation in a sphere, so we shall use spherical coordinates (r, , ). First, we translate our boundary condition. Note that in spherical coordinates g(
Solutions to Homework # 4 Math 381, Rice University, Fall 2003 Problem 1: We start by making the Euler substitution t = ln x, or x = et . Then we compute that dy = xy , dt and d2 y = x2 y + xy . dt2
So we can rewrite our original equation as dy d2 y + ( -
HOMEWORK ASSIGNMENT # 4
MATH 211, FALL 2006, WILLIAMS COLLEGE
Abstract. These are the instructors solutions.
1. Problem: Cofactors and Cramers rule
(1) Use the classical adjoint method
2
A = 4
6
(2) Use Cramers rule to
x1
x1
x1
to compute the inverse of
HOMEWORK ASSIGNMENT # 1
MATH 211, FALL 2006, WILLIAMS COLLEGE
Abstract. These are the instructors solutions to the rst homework.
1. Problem One
Describe, in your own words, the geometric way to multiply complex
numbers. Draw pictures and explain in full s
Second Exam
Math 211, Williams College
November 10, 2006
Directions: This Exam has 4 questions. Be sure to show
all of your work and explain yourself carefully. You have
two and a half hours to complete this exam, to be done
in a single block of time. You
SECOND EXAM SOLUTIONS
MATH 211, WILLIAMS COLLEGE, FALL 2006
Abstract. These are the instructors solutions for the second exam. For
statements of the problems, see the posted copy of the exam.
1. Problem One
To see if W1 and W2 are the same, we apply the r
Final Exam
Math 211, Williams College
December 15, 2006
Print your name here:
Directions: This Exam has 6 questions on 8 pages, including this cover. Be sure to show all of your work. You
have two and a half hours to complete this exam. Please
sign the ho
HOMEWORK ASSIGNMENT SOLUTIONS# 2
MATH 211, FALL 2006, WILLIAMS COLLEGE
Abstract. These are the instructors solutions for assignment 2.
1. Problem One
Use the elimination/back-solving algorithm to solve the following
systems. (i.e. put the system in row ec
HOMEWORK ASSIGNMENT # 3
MATH 211, FALL 2006, WILLIAMS COLLEGE
Abstract. These are the instructors solutions.
1. Problem: On LU decompositions
This problem will lead you through understanding the LU decomposition
of a matrix. Your answer for this problem s