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 = (
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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
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 O
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 t
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 o
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 o
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 clea
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,
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. Moder
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
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 adv
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 the
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
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 Ei
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 c
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. S
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 ru
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. Alw
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
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
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
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 s
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 func
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
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.
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
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 t