Algebra Workshop 2: Factoring polynomials
1
The standard expressions
The three formulae you have to know: for any expression E and F,
E 2 + 2EF + F 2 = (E + F )2
E 2 2EF + F 2 = (E F )2
E 2 F 2 = (E F )(E + F )
Examples:
x2 + 6x + 9 = (x + 3)2
x2 + 2
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AMS 15A, Midterm 2
Name: _ _
You will be allowed to use electronic items to check your answers for the last 25 minutes of the midterm.
ALL YOUR ANSWERS MUST BE FULLY J USTIFIED. WOLFRAM ONLY ANSWERS WILL ONLY
GET YO
AMS 15A, Midterm 2
Name:
You will be allowed to use electronic items to check your answers for the last 25 minutes of the midterm.
ALL YOUR ANSWERS MUST BE FULLY JUSTIFIED. WOLFRAM ONLY ANSWERS WILL ONLY
GET YOU A MINIMAL AMOUNT OF POINTS.
Read all the qu
AMS 15Aa Midterm 1
Name: _' _ _
Electronic items are not allowed.
Read all the questions before you start working on any of them. Start with the ones you are most
comfortable with, and continue with the other ones later. Always double-check your answers.
AMS 15A, Practice for Midterm 2
Name:
You will be allowed to use electronic items to check your answers for the last 15 minutes of the midterm.
Read all the questions before you start working on any of them. Start with the ones you are most
comfortable wi
AMS 15A, Practice for Midterm 2
Name:
You will be allowed to use electronic items to check your answers for the last 15 minutes of the midterm.
Read all the questions before you start working on any of them. Start with the ones you are most
comfortable wi
76
CHAPTER 2. TOOLS FOR DERIVATIVES
We see that the two planets regularly come very close to one another. but the period of these events
is neither 365 days, nor 687 days. What is it? Also, it is clearly much easier to y a spacecraft to Mars
if the two pl
85
o
We nish this long chapter by introducing a nal set of functions, and their derivatives: the inverse
trigonometric functions. They have a number of applications, one of which we will study today, related
to F1 car races.
2.10
Inverse trigonometric fun
63
2.5
2.5.1
The derivative of the exponential function
Case Study: The IQ test
IQ tests are standardized tests designed in the late 19th and early 20th century to assess a persons intelligence. Modern IQ tests measure several factors which are related to
42
CHAPTER 2. INTRODUCTION TO DERIVATIVES
In the previous Section, we learned about derivatives, and the relationship between derivatives and
properties of their graphs. Unfortunately, the method for calculating these derivatives was really slow and
cumbe
47
In the previous Section, we learned a rst set of tools for derivatives, including derivatives of sums
and dierences of functions, and derivatives of power functions. Here, we continue with more advanced
tools including the derivatives of products and q
70
CHAPTER 2. TOOLS FOR DERIVATIVES
There is one nal family of functions we have not studied yet, which is the family of trigonometric
functions: sine, cosine, tangent, and all the ones related to them: cotangent, secant, cosecant, etc. These
functions ha
92
CHAPTER 3. LIMITS
Chapter 3
Limits
In many of the previous lectures, we have been using the concept of limits to express our interest in
nding out what happens to f (x) when x tends to 0?, or what is the behavior of f (x) when x tends
to innity?. In th
104
CHAPTER 3. LIMITS
3.2
Limits at a point and the notion of continuity
3.2.1
Case study: There, and back again (special relativity style).
The theory of special relativity was developed by Albert Einstein in 1905, and studies what would happen
if we tri
80
2.9
2.9.1
CHAPTER 2. TOOLS FOR DERIVATIVES
Derivatives of inverse functions and logarithms
Case study: Lets go to Mars! (part V)
Now that we know the Earth-Mars distance as a function of time, we can also try to predict how Mars
luminosity, as seen fro
AMS 15A Practice Final
TA: _ _
Section: _ _.
March 15, 2012
Non-graphing scientic calculators are allowed but should not be necessary.
Read all the questions before you start working on any of them. Start with the ones you are most
comfortable with, and c
113
In this nal lecture of the quarter, we will use a number of techniques we have learned so far to study
a rather important real-life application, i.e. banking interest rates.
3.3
3.3.1
LHopitals rule
Case study: Interest rates
In banking, interests are
AMS 15A Final
Name: _ _
March 21, 2012
a:
Read all the questions before you start Working on any of them. Start with the ones you are most
comfortable with, and continue with the other ones later. Always double-check your answers.
Relax, and good luck!
GR
Handout: Formulary for derivatives
1
Rules for derivatives
In the following, the functions f (x) and g(x) are assumed to be dierentiable. a is a constant.
Addition rule:
[f (x) + g(x)] = f (x) + g (x)
(1)
Linearity rule:
[af (x)] = af (x)
(2)
Product r
56
CHAPTER 2. TOOLS FOR DERIVATIVES
We now continue on our exploration of tools for derivatives with a case study that introduces the
Chain Rule.
2.4.6
Case Study: Computer-generated graphics: how to cast a shadow.
Today, most computer-games, movies and a
11
Generally speaking, we will nd that data on its own is never sucient. In order to be able to use the
data in practice, the question will always be What is the best function to t this data? In this Section,
we will learn to recognize some of the basic f
22
CHAPTER 1. FUNCTIONS
In the previous lectures we studied linear, exponential, power and logarithmic functions. We now
continue by looking at polynomial and rational functions. These functions are some of the most commonly
used in Applied Mathematics, a
Chapter 1
Functions
In this Chapter, we will learn about mathematical functions, and in particular:
Why is the concept of functions so useful?
Why is it so important to know what kinds of basic functions exist, and what their properties are?
We will loo
Algebra Workshop 4: Expressions with exponents
1
Rules for expressions with exponents
Review the following rules:
am an = am+n and am an = amn
(am )n = amn = (an )m
am /an = ann
(ab)m = am bm
a m
b
=
am =
am
bm
1
am
a0 = 1
a m
b
b m
a
=
Note: These
63
2.5 The derivative of the exponential function
2.5.1 Case Study: The IQ test
l.
I Q tests are standardized tests designed in the late 19th and early 20th century to assess a persons in
telligence. _Modern IQ tests measure several factors which are rela
Algebra Workshop 1: Simple manipulation of expressions
1
Manipulating negative signs
1.1
Removing brackets in an expression
Rule:
Brackets preceded by a + sign can just be removed
Brackets preceded by a sign required some care: think of them as 1 (expre
32
CHAPTER 1. FUNCTIONS
1.4.6
Mathematical Corner: Rational functions
General properties
Definition:
Domain of definition:
Examples:
Asymptotic behavior of rational functions
In order to nd out what the shape of the graph of a rational function may look l