MAT 132 MidTerm Exam
2009 Summer II
Name:
ID:
Problem 1 2 3 4 5 Total
Points 20 10 10 10 10
60
You get
Instruction: You have 1 1 hour to nish this exam. The total pages of this
2
exam (including this page) is 8. Calculator or any kind electronic device is
NOTES ON SECOND ORDER LINEAR DIFFERENTIAL EQUATIONS
Contents
1.
2.
3.
4.
5.
6.
7.
8.
Generalities
Characteristic equation with dierent roots
Characteristic equation with repeated roots
Characteristic equation with no real roots
Summary on solving the line
MAT132 PAPER HOMEWORK 5
DUE IN RECITATION ON 4/30 OR 5/1
Problem 1. We will prove the Generalized Binomial Theorem; that is, for any k and for |x| < 1
(1 + x)k = 1 + kx + +
(i) Show that
(ii) Let g(x) =
k1
n
+
k
n=0 n
k1
n1
=
k(k 1) (k (n 1) n
k n
x + =
x
Question 1.
a) Use the trapezoidal rule with n = 6 to approximate
sin x dx
0
b) According to the error bounds formula, what is the maximum error in the above approximation?
c) Sketch the region S = cfw_(x, y) | x 1, 0 y ex . Express the area of S as an im
Question 1.
a) Use the trapezoidal rule with n = 6 to approximate
sin x dx
0
b) According to the error bounds formula, what is the maximum error in the above approximation?
c) Sketch the region S = cfw_(x, y) | x 1, 0 y ex . Express the area of S as an im
Here are some practice questions for the rst midterm. Some questions may be useful on the
midterm, some may not. . . Indeed, one of the (sub)-questions will appear on the midterm itself!
Question 1. Let
(x + 1)2 + 1 for 2 x 0
f (x) = (x 1)2 1
for 0 x 2
0
MAT 132 Calculus II
Final Exam Topics
This is the description of problems that may appear on the nal exam. (The actual
practice questions can be found in the textbook or past homework or exams.) The goal of
this list is to help you to focus on essential s
MAT 132 Practice Exam
2009 Summer II
1. Evaluate the integrals.
(a)
1
2
xex dx
1
(b)
x sin x dx
(c)
2
x2 + 5x + 6
dx
x
1
2. Find the derivative of
x3
sin t dt.
x2
3. Determine whether the improper integral is convergent or divergent. If
convergent, please
Notes on Second Order Linear Differential Equations
Stony Brook University Mathematics Department
1. The general second order homogeneous linear differential equation with constant coefcients
looks like
Ay + By +Cy = 0,
where y is an unknown function of t