Math 339 Midterm Review
(1) Given the matrix
1 0 0
A = 0 1 1 .
0 1 1
(a) Find the eigenvalues and eigenvectors.
(b) Diagonalize A, that is, find matrices P and D such that A =
PDP1 .
(c) Compute A7 using the decomposition in part b.
(2) (a) Find the LU de
Math 339 Quiz 3 Redo
Let
e1 =
3
0
, e2 =
0
1
, y1 =
2
5
, y2 =
1
6
and let T : R2 R2 be a linear transformation that maps e1 into y1 and
maps e2 into y2 . Find the images of
7
x1
and
.
2
x2
1
MA 180* Fall 2010* Exam 1
Name _
_ of 80 points
There are twenty questions on this exam worth four points each. Partially correct answers may receive
partial credit. Put all answers in the answer column. Only the answers in the box will be graded. Use
scr
MA132: Calculus II Exam 2 (14 March 2002)
Name: Student Number:
(12) Problem 1. For the region enclosed by the curves y 2 = 4x + 16 and 2x + y = 4: (a) Sketch the region (label intersection points).
t (0, 4) 6 A
y
ANSWER: To plot rst curve (quadratic), nd
MA132: Calculus II Exam 3 (18 April 2002)
Name: Student Number:
(20) Problem 1. Test each series for convergence. Show your work and cite the test(s) used. (a) 8n n=0 (n + 1)!
ANSWER: Use the Ratio Test: an+1 an 8n+1 8n+1 (n + 1)! 8 (n + 2)! = = = 0=L<1 n
MA132: Calculus II Exam 3 (18 April 2002)
Name: Student Number:
(20) Problem 1. Test each series for convergence. Show your work and cite the test(s) used. (a) 8n n=0 (n + 1)! 1 2 n=2 n [ln(n)]
(b)
(20) Problem 2. Test each series for convergence. Show
MA132-Calculus II Exam 2 13 June 2006
Instructions: Show your work.
Name: Student Number:
Answers without sufficient justification may not receive full credit. No books, notes, or calculators. Time limit: 90 minutes. Some integrals which might help: sec(
MA132Calculus II Exam 2 (13 June 2006)
Name: Student Number:
(12) Problem 1. Consider the region between the curves y = 2 x and y = x2 for 0 x 2. (a) Sketch the region. Label the axis scales and any points of intersection. ANSWER: y 6 4 3 2
d d
1
d d d d
MA132 Exam 1 2 June 2006
Instructions: Show your work.
Name: Student Number:
Answers without sucient justication may not receive full credit. No books, notes, or calculators. Time limit: 90 minutes. Some integrals which might help: sec(x) dx = ln |sec(x)
MA132: Calculus II Exam 2 (14 March 2002)
Name: Student Number:
(12) Problem 1. For the region enclosed by the curves y 2 = 4x + 16 and 2x + y = 4: (a) Sketch the region (label intersection points). (b) Write an integral which represents the area of the r
MA132: Calculus II Exam 1 (14 February 2002)
(10) Problem 1. Find the integral: x cos(3x) dx.
Name: Student Number:
ANSWER: Using integration by parts with u = x and dv = cos(3x) dx: 1 1 x cos(3x) dx = x sin(3x) 3 3 1 1 sin(3x) dx = x sin(3x) + cos(3x) +
MA132: Calculus II Exam 1 (14 February 2002)
(10) Problem 1. Find the integral: (8) Problem 2. Find the integral: (8) Problem 3. Find the integral: (12) Problem 4. Find the integral: (12) Problem 5. Find the integral:
Name: Student Number:
x cos(3x) dx. [
MA 181 Lecture
Chapter 10
College Algebra and Calculus by Larson/Hodgkins
Exponential and Logarithmic Functions
10.5) Derivatives of Logarithmic Functions
You can find a review of exponential functions in Sections 10.3.
We can use implicit differentiation
MA 181 Lecture
Chapter 11
College Algebra and Calculus by Larson/Hodgkins
Integration and Its Application
11.2) Integration by Substitution and the General Power Rule
Recently we used the Simple Power Rule to do some basic antidifferentiation.
n
x dx
x
MA 181 Lecture
Chapter 11
College Algebra and Calculus by Larson/Hodgkins
Integration and Its Application
11.1) Antiderivatives and Indefinite Integrals
This chapter begins a new process called antidifferentiation that is considered the inverse applicatio
MA 181 Lecture
Chapter 8
College Algebra and Calculus by Larson/Hodgkins
Applications of the Derivative
8.4) Increasing and Decreasing Functions
A function is increasing if its graph moves up as x moves to the right and is decreasing if its graph
moves do
MA 181 Lecture
Chapter 10
College Algebra and Calculus by Larson/Hodgkins
Exponential and Logarithmic Functions
10.3) Derivatives of Exponential Functions
You can find a review of exponential functions in Sections 10.1 and 10.2.
Definition of the Number e
MA 181 Lecture
Chapter 9
College Algebra and Calculus by Larson/Hodgkins
Further Applications of the Derivative
9.2) Business and Economics Applications
In this section we expand the concept of optimization to business applications.
Example:
A company has
MA 181 Lecture
Chapter 8
College Algebra and Calculus by Larson/Hodgkins
Applications of the Derivative
8.3) Related Rates
First we give special attention to notation. When we say
dy
d
or
, we are saying the derivative with
dx
dx
respect to the variable x
MA 181 Lecture
Chapter 8
College Algebra and Calculus by Larson/Hodgkins
Applications of the Derivative
8.2) Implicit Differentiation
We say that a function is in explicit form if it is of the form y=f(x). In other words, on variable is
explicitly defined
MA 181 Lecture
Chapter 8
College Algebra and Calculus by Larson/Hodgkins
Applications of the Derivative
8.1) Higher Order Derivatives
The derivative of f is the second derivative of f and is denoted f.
Equivalently,
d
f ' ( x) f ' ' ( x)
dx
Similarly, th
MA 181 Lecture
Chapter 9
College Algebra and Calculus by Larson/Hodgkins
Further Applications of the Derivative
9.1) Optimization
One of the most important applications of the derivative is optimization. It involves finding a value of x
where we can eithe
MA 180 Lecture
Chapter 2
College Algebra and Calculus by Larson/Hodgkins
Functions and Graphs
2.1) Lines in the Plane
Definition of the Slope of a line
The slope m of the nonvertical line passing through the points x1 , y1 and x2 , y2 is m
y2 y1
x2 x1
wh
MA 180 Lecture
Chapter 2
College Algebra and Calculus by Larson/Hodgkins
Functions and Graphs
2.1) Graphs of Equations
We represent real numbers on a real number line in a one-dimensional representation. We can
represent a two dimensional representation i
MA 180 Lecture
Chapter 1
College Algebra and Calculus by Larson/Hodgkins
Equations and Inequalities
1.6) Linear Inequalities
Simple inequalities are used to order real numbers. To solve an inequality in the variable x we find all
values of x for which the
MA 180 Lecture
Chapter 1
College Algebra and Calculus by Larson/Hodgkins
Equations and Inequalities
1.7) Other Types of Inequalities
In this section it is important to remember that whenever we multiply or divide by a negative number we
must switch the in
MA 180 Lecture
Chapter 1
College Algebra and Calculus by Larson/Hodgkins
Fundamental Concepts of Algebra
1.1) Linear Equations
Equations and Solutions
An equation is a statement that two algebraic expressions are equal. Examples include 3x 4 5x 2
and
x 1
MA 180 Lecture
Chapter 1
College Algebra and Calculus by Larson/Hodgkins
Equations and Inequalities
1.3) Quadratic Equations
Solving Quadratic Equations by Factoring
Definition of a Quadratic Equation
A quadratic equation in x is an equation that can be w