Linear Algebra Cheat Sheet
by Matt Hanson
A note on organization: topics are presented first with respect to matrix
operations in part I, and then with respct to the more general form of vector
spaces in part II.
PART I: Matrix Operations
Intersect:
The i
Proof of the General Taylor Theorem
March 9, 2017
Statement of the theorem
Given some function, f (x) defined at the value x = c for some c 2 R, with f (x)
infinitely dierentiable, we may write:
f (x) =
1
X
(x
n=1
c)n dn f (x)
|x=a
n!
dxn
where the series
Proof of the Fundamental Theorem
of Calculus
Proof of the First Part:
First, we define a function, for some a 2 R,
x
F (x)
f (t)dt
a
Where F (x) and f (x) are continuous and dierentible on [a, b] and (a, b), resepctively. Then for some 2 (a, b), we may
Calculus II cheat sheet:
by Matt Hanson
Note: all dierential equations topics are covered in a dierent document.
Hyperbolic functions and properties:
et + e
2
t
et
e
t
et e
tanh(t) = t
e +e
t
cosh(t) =
sinh(t) =
=)
d
cosh(t) = sinh(t) =)
dt
cosh(t)dt = si
Calculus Topics
1. Sequences
A. Limit proofs
B. Evaluate limits (if they exist) using
1. factoring
2. rationalization
3. do algebra
II. Limits of Functions
A. Epsilon/Delta limit proofs
B. One-sided limits and existence of limits
C. Finding limits/one-sid
Calculus I Assignment 5.
Due: No later than 3pm on Thursday, October 15th.
(1) Evaluate limx0 x3 cos x1 .
(2) Evaluate limx2+ (x + 1) + (x 2)2 sin (sec (/x)
(3) Suppose, for all x in some open interval containing x = 0, that
3|x| + 5|x 5| f (x) 6|x| + (x
Data Given:
Population of the U.S. from 1790 1950.
Year
1790
1800
1810
1820
1830
1840
1850
1860
1870
1880
1890
1900
1910
1920
1930
1940
1950
Population
3,929,000
5,308,000
7,240,000
9,638,000
12,866,000
17,069,000
23,192,000
31,443,000
38,558,000
50,156,0
MTH 201 Applied Mathematics
Chapter 6 - Sample Quiz Exercises
Name (Print):_Show Work
_
1. Let = cfw_ | is an even integer and 0 10 = cfw_ | is an integer and 2 5 and
= cfw_1 1 3 5 9 10 Mark the following statements as true/false.
(a) =
(i) True
(ii) Fa
Worksheet: Chapter 2 Section 2: Answers
1. Sketch the curves y = x3 3 and y = 2. Calculate the area bounded between the curves from x = 0 to x = 1.
Area:
5(0)
4
0
4
=
h
1
1
2 x3 3 dx = 0 5 x3 dx = 5x
0
x4
4
i1
= 5(1)
0
14
4
19
4
2. Sketch the curves y
Worksheet: Section 3.4: Limits at Points of Possible Discontinuity: ANSWERS
1.
lim 2x3
x3/2 4x6
2(3/2)3
Evaluation at x = 3/2: 4(3/2)6
= 00 .
Divide out (x 3/2) or , using the denominator of the fraction 3/2, divide out 2(x 3/2) = 2x 3.
numerator: 2x3
2x3
Worksheet: Section 2.3: Integral EstimatesReimann Sums : ANSWERS
0
1. Determine the Riemann Sum for 4 x2 + 1 dx using n = 4 subintervals. Use evaluation at the right end of
the subintervals, then evaluation at the left end of the subinterval. Finally, giv
Worksheet: Section 3.1: Limits from graphs: ANSWERS
1. The following represents a graph of a function y = g(x).
Find the following limits.
(a) lim g(x) = 3.5
x3
(b) lim+ g(x) = 6
x3
(c) lim g(x), does not exist
x3
(d) g(3) = 3.5
(e) lim+ g(x) = 1
x0
(f) l
Worksheet: Section 3.3: Concepts of Infinity and Division by Zero: ANSWERS
For the following (conceptual) forms, state whether they are determinate or indeterminate. In the case they are
determinate, reduce the form to 0 , , or .
1. 4 + =
2. 4 =
3. + =
Worksheet: Chapter 5 Section 2: Extrema and the First Derivative Test:
ANSWERS
2
1. Find any relative extrema (x, y) for the function f (x) = x2 3x.
f 0 (x) = x 3, 0 = x 3, x = 3. There is one critical value: x = 3. Test f 0 (x) at x = 0 and x = 4.
f 0 (0
EXPLAINING ADDITIVE AND MULTIPLICATIVE
INVERSES
JORDAN DUBUQUE
ADDITIVE INVERSE PROPERTY:
The additive inverse of a number is the same number with the opposite sign. When adding
a number to its additive inverse, the sum will always be 0. This is the addi
HISTORY OF LIFE
LAB
Jordan Dubuque
October 4, 2016
When and where did
cyanobacteria life begin and how
do
we
know
this?
The oldest evidence of life belongs to microscopic marine
organisms during the Precambrian era (roughly 3.5 million years
old).
Fossil
Jordan Dubuque
Journal: Cancer Cells
There are more than 200 different types of cancer. 1 in 2 people in the UK will get cancer in their
lifetime. Cancer starts when gene changes make one cell or a few cells begin to grow and multiply too
much. This may c