CHAPTER
1
First-Order Differential Equations
1.1
! 1.
Dynamical Systems: Modeling
State Variables Temperature, acidity, how much is in the glass, percent of glass full, color, maybe even taste if
MAT 116 : Midterm Exam Solutions
Last Updated: July 14, 2010
For each problem, the first solution presented is the instructors most anticipated solution for students
to give. Any subsequent solutions
MAT 116 In-class Problems (#7)
July 20, 2010 and July 21, 2010
Let Z+ denote the set of nonnegative integers; if n N, let nZ+ denote the nonnegative
multiples of n. Given k N and n Z+ , let k,n denote
MAT 116 In-class Problems (#4)
June 30, 2010 and July 1, 2010
Pidgeonhole Principle (Simple Version). If n + 1 objects are distributed into
n boxes, then at least one box contains two or more of the o
MAT 116 In-class Problems (#3)
June 28, 2010 and June 29, 2010
1. Consider the multiset S = cfw_2 a, 1 b, 3 c of six objects of three types.
(a) Find the number of permutations of S.
(b) Find the numb
MAT 116 In-class Problems (#8)
July 22, 2010 and July 26, 2010
Problem 1. Find the number hn of bags of (n pieces of) fruit that can be made out
of apples, bananas, oranges, and pears, where, in each
MAT 116 In-class Problems (#6)
July 12, 2010 and July 13, 2010
The Inclusion-Exclusion Principle.
Let S be a finite set. Suppose that A1 , A2 , . . . , Am are subsets
of S. Then
|A1 A2 Am | = |S| |Ai
MAT 116 In-class Problems (#1)
June 21, 2010
1. How many 2digit numbers are there?
2. How many ways are there to form a 3letter word (i.e. sequence) using the
letters A, B, C, D
(a) with repetitions o
MAT 116 In-class Problems (#2)
June 22, 2010 and June 23, 2010
1. How many distinct positive divisors does each of the following numbers
have?
(a) 34 52 76 11
(b) 620
(c) 1010
2. What is the number of
MAT 116 In-class Problems (#5)
July 6, 2010
Remark. Just to be clear, when we say the box contains r objects, we really mean
that the box contains at least r objects.
Pidgeonhole Principle (Strong For
304
CHAPTER 4
Second-Order Linear Differential Equations
4.4
! 1. 3. 5. 7. !
Undetermined Coefficients
Inspection First y - y = t y p (t ) = -t
y = 2 y p (t ) = t 2
2. 4. 6. 8.
y + y = 2
CHAPTER
4
Second-Order Linear Differential Equations
4.1
! 1.
The Harmonic Oscillator
The Undamped Oscillator ! ! + x = 0 , x(0) = 1 , x(0) = 0 x
x The general solution of the harmonic oscillator
MAT 116 : Midterm Exam Study Guide
Michael Williams
Last Updated: July 1, 2010
Remarks
The midterm exam will consist mostly of computations, and it might also contain some short proofs
or explaination