00
Problem 1. Compute 199 .
A) 1
B) 99
C) 100
D) 100!
E) None of these
Problem 2. For any positive integers n, a, and b with 0 < a < b < n we have
n
n
a < b.
A) True
B) False
Problem 3. Suppose that S is a set with n elements and R is an equivalence
relat
Integration By Parts
The Tabular Method
Sometimes it is necessary to apply the method of integration by parts several times before we obtain
an integral we can evaluate. In the case where the integral is of the form
p(x)f (x) dx
where p(x) is a polynomial
Integration By Parts
The Product Rule
Let f and g be functions and suppose that G is an antiderivative for g . This means that G (x) = g (x).
The product rule states that
d
[f (x)G(x)] = f (x)g (x) + f (x)G(x).
dx
We can rewrite this rule in integral nota
Hyperbolic Functions
Definition
We can express the exponential function ex as the sum of an odd function and an even function:
ex + ex
ex ex
+
.
2
2
The even part of ex is called hyperbolic cosine and we denote it by
ex =
Definition
cosh x =
ex + ex
.
2
T
Length of a Plane Curve
Arc Length
Definition Let f be a dierentiable with a continuous derivative f . In this case f is called a
smooth function and the graph y = f (x) is called a smooth curve.
We wish to have a notion of the length along a smooth curve
Functions and Integration
Functions Defined by Integrals
Let f be a continuous function. Then by the Fundamental Theorem of Calculus, the function
x
g (x) =
f (t) dt
a
is an antiderivative for f (x). In some cases it is advantageous to dene a function g i
Math 2001 Homework #13 Solutions
1. Section 7.1: #3. For each part below, draw a graph with the described properties.
(a) A graph with three vertices and four edges.
Solution: If we have vertices cfw_1, 2, 3, there is no simple graph that has four
edges (
Math 2001 Homework #8 Solutions
1. Section 3.2: 11. Suppose a, b, q , and r are nonzero integers such that a = bq + r. Prove
or disprove each of the following statements.
(a) gcd(a, b) = gcd(b, r)
Solution: Importantly, note that we cannot use the Euclide
Math 2001 Homework #7 Solutions
1. Section 3.1 #2. Let a, b, c, d be nonzero integers. Prove the following implications.
(a) If a|b and a|c, then a2 |bc.
Solution: Suppose a|b and a|c. Then there are integers j and k such that b = aj
and c = ak . Therefor
Math 2001 Homework #9 Solutions
1. (a) Roughly how many digits does the number 3893 have?
Solution: Very roughly, we use the fact that 32 = 9 10 and get 3893 3 10446 ,
so we get something slightly less than 447 digits. More precisely if I plug into
Google
Math 2001 Homework #10 Solutions
1. Section 4.1: 6ab. For each map below, determine the number of southerly paths from
point A to point B .
Solution: We just have to use the same process as we did in building Pascals triangle:
mark a 1 to count the paths
Math 2001 Homework #11 Solutions
1. Section 4.5: #7. An integer between 100 and 999 inclusive is selected at random. If
n = 100d2 + 10d1 + d0 has digits d2 , d1 , and d0 , nd the probability that:
(a) The digits of n are all distinct.
Solution: Since 100
Math 2001 Homework #12 Solutions
1. Section 6.1: #2bcd. Determine whether the following relations on cfw_1, 2, 3, 4 are functions on cfw_1, 2, 3, 4. For each function, determine whether it is one-to-one, determine
whether it is onto, and give its range.
(
Math 2001 Homework #6 Solutions
1. Section 2.1 #11. Suppose m and n are integers. Prove that the following statements
are equivalent.
(a) m2 n2 is even.
(b) m n is even.
(c) m2 + n2 is even.
Solution: We need three cyclic proofs: we can either do (a)(b),
Trig Integrals
Products of Sines and Cosines
We can sometimes integrate a function of the form sinm x cosn x using an appropriate u-substitution.
We can u-substitution if n or m is odd. The appropriate choice of u is given in the following table:
Evalutat
Trig Substitutions
Eliminating Radicals
Our goal is to use certain u-substitutions involving trig functions to eliminate the radicals inside an
integral that contains one of the expressions:
a2 x2
x2 a2
or
or
The appropriate substitutions are as follows:
Method of Partial Fractions
Partial Fraction Decomposition
Suppose we have a rational function, i.e. a function of the form R(x) = P (x)/Q(x) where P and
Q are polynomials. Under certain conditions we can decompose R as a sum of simpler rational
functions
Problem 1. Let S be a set with three subsets A, B , and C . Suppose that x is
an element of S . How many of the sets
A, B, C, A B, A C, B C, A B C
could contain x? Answer as precisely as possible.
A) 1
B) 7
C) an even number
D) an odd number
E) Any number
Problem 1. Let n be a positive integer. Compute
n
(1)k
k=0
n
k
=
n
n
n
n
+
+ (1)n
.
0
1
2
n
Problem 2. Let S be a set with n elements. How many ways are there to
partition S into two subsets.
A) 1
B) n
C) 2n 2
D) 2n
E) None of these
Problem 3. Let S be a
Problem 1. Is cfw_1, cfw_2, 3, cfw_5 a partition?
A) Yes
B) No
C) Of what?
Problem 2. How many partitions are there of the empty set?
A) 0
B) 1
C)
D) The answer is not dened.
Problem 3. How many distinct rearrangements are there of the letters of my
name
Problem 1. How many of the integers x with 1 x 120 are divisible by at
least one of 2 or 5?
A) 12
B) 60
C) 72
D) 84
E) 96
Problem 2. How many of the integers x with 1 x 120 are divisible by at
least one of 2, 3, or 5?
A) 4
B) 40
C) 76
D) 108
E) 112
Proble
Problem 1. Prove by contradiction that the sum of a rational number and an
irrational number cannot be rational.
Problem 2. How many ways are there to rearrange the list (1, 2, 3, 4, 5) such
that each of 1, 3, and 5 does not wind up in the same place?
A)
Problem 1. Let n be a positive integer. Compute
n
(1)k
k=0
n
k
=
n
n
n
n
+
+ (1)n
.
0
1
2
n
Solution. The answer is 0. One way to see this is to remember that (x + y )n =
n
n k nk
and substitute x = 1 and y = 1.
k=0 k x y
Here is a combinatorial proof. R
Problem 1. Is cfw_1, cfw_2, 3, cfw_5 a partition?
A) Yes
B) No
C) Of what?
Problem 2. How many partitions are there of the empty set?
A) 0
B) 1
C)
D) The answer is not dened.
Solution. The empty partition is the only one.
Problem 3. How many distinct rea
00
Problem 1. Compute 199 .
A) 1
B) 99
C) 100
D) 100!
E) None of these
Problem 2. For any positive integers n, a, and b with 0 < a < b < n we have
n
n
a < b.
A) True
B) False
Problem 3. Suppose that S is a set with n elements and R is an equivalence
relat
Problem 1. Draw a picture of congruence modulo 4 as a relation on the integers.
Problem 2. Every relation on the empty set is an equivalence relation.
A) True
B) False
Problem 3. Suppose that Q and R are equivalence relations on a set S . Is
Q R an equiva
Differential Equations Summary
We give a summary of our results about dierential equations in the form of a ow chart. We will
assume that rst order dierential equations are either linear or separable and that second order
dierential equations are homogene
First Order Differential Equations
A dierential equation is an equation involving one or more derivatives of an unknown function.
A function y (x) is a solution of a dierential equation if the equation is satised when y and its
derivatives are substituted
Improper Integrals
We can extend the notion of the denite integral (in which we usually require functions to be
continuous and intervals to be nite) in two ways. One way is to allow integrals over innite
intervals. The other is to allow the integration of
Using Integral Tables
For this section, refer to the integral table on the back inside cover of the text. Some common
integrals can also be found at http:/www.integral-table.com/
Perfect Matches
We have discussed several methods of integration so far, but