Section 5.2: The Definite Integral
Def
Math 20
Let f be a function defined on [ a, b ] where [ a, b ] is divided into n subintervals of equal width. Let
x1 , x2 ,.xn be any sample points in the subintervals. Then the definite integral of f from a to b is
Section 5.3: The Fundamental Theorem of Calculus
Suppose f is a continuous function on [ a, b ] . Define a new function for area.
Math 20
g ( x) = f (t )dt
x
a
Example
Show f ( x) = x on [1, 4]
Fundamental Theorem of Calculus Part I
Suppose f is continuou
Section 5.5: The Substitution Rule
If f ( x) = e x , what is f ( x) ?
2
Math 20
f ( x) = _. Thus
But how do we do this from the integral side? We can use the _ method.
2
2 xe x dx
How to choose u ( x) : Pick the piece of the integral that you have the der
Section 5.1: Areas and Distance
Math 20
Example Find the area under the curve of y =
2
x from 0 x 3 .
3
Area =
Example Find the area under the curve of y = x 2 from 0 x 1 .
For this function we will need to
_ the area.
We _ the interval [ 0,1] into
four _
Section 4.1: Maximum and Minimum Values
Math 20
Def
A function with domain D has an absolute (or global) maximum at c if f ( c ) f ( x ) for all x in
Def
A function with domain D has an absolute (or global) minimum at c if f ( c ) f ( x ) for all x in
D.
Rolles Theorem Let f be a function such that
1) f is continuous on a, b .
2) f is differentiable on a, b .
3) f a
f b
In this case, there is a number c in a, b with f c
0.
Proof:
If f is a constant, then c can be any number in a, b .
If _ for some x in a,
Section 5.4: Indefinite Integrals and the Net Change Theorem
Math 20
Def An antiderivative of f is denoted
This is called the _ integral.
f ( x)dx
Note:
1)
f ( x)dx
is a _ whose derivative is _.
b
2)
is a _ with various interpretations: _
a
f ( x)dx =
Written Homework 5 Math 330 Fall Name: [/qu cfw_35533. RQJL'ILS
This assignment is based on lessons 1720.
1) You want to write a function, x], to represent the money in an inheritance
account. Every year since the account was started [it was started in 20
Written Homework 1 Fall Math 330 Name:
% meme:
This written homework is based on lessons 14.
1. Simplify the following: You must Show your work in a neat and orderly
fashion and do not skip steps. I want you to Show each equivalent expression
below the pr
Section 3.1: Derivatives of Polynomials and Exponential Functions
Math 20
Properties:
1) If f ( x) = c and c is a constant, then f '( x) = _ .
2) If f ( x) = x then f '( x) = _ , or
d
( x ) = _ .
dx
Try to find the pattern.
If f ( x) = x 2 , then f '( x)
Section 1.1: Four Ways to Represent a Function
Math 20
Question: What is Calculus?
Answer: The study of how things _.
There are two basic areas:
1) Velocity -
Recall: Distance = _
a)
b)
This graph shows a _ rate/slope.
But what is the rate/slope of this g
Section 5.1: Areas and Distance
Math 20
Example Find the area under the curve of y =
2
x from 0 x 3 .
3
Area =
Example Find the area under the curve of y = x 2 from 0 x 1 .
For this function we will need to
_ the area.
We _ the interval [ 0,1] into
four _
Section 4.1: Maximum and Minimum Values
Def
Math 20
A function with domain D has an absolute (or global) maximum at c if f ( c ) f ( x ) for all x in
D. f ( c ) is called a maximum value of f on D.
Def
A function with domain D has an absolute (or global)
Section 2.1: The Tangent and Velocity Problems
Math 20
Tangent From the Latin word that means touching
_
_
Example
Find the equation of the tangent line to y = sin x at x = 1 .
How can we get a better approximating line? _. Find one such line.
Actual tang
9
3
1. Simplify 27 x x 2 y 6
3
6 +2 x
2. If f ( x )= 2
, then find
x + x20
lunches and dinners maximize her
f (5)
and find the domain of
f ( x) .
3. Solve
y =10
cfw_43 x
x+ 9 y=0
cfw_
x+ 4 y=2
5
x y + z=15
4. Solve
3 x2 y + z=12
5. Solve and graph
10 x+52
1. Solve 5 ( x2 )+ 2 x=9x
2. Graph x3 y6=0
3. Graph 5 x+ 9 y =10
5 x2 y
4. Simplify
2
( 5 x 2 ) y2
5. What is the equation of the line that
cfw_
2
y x+1
3
2 x+ 4 y >8
17. It takes me 2 hours of cutting and 4
16. Solve with a graph
hours of sewing to make
1. Make a lovely graph of
y = x^2 + 8x + 20
2. Make a cool graph of
3.
4.
5.
6.
7.
8.
y = 2x^2 + 5x 1
5
3
2x
= 2
Solve
x +2 x2 x 4
3 4
=x
Solve
x 3
Solve x 21 x=1
Solve Solve x 2+ 10=6 x
Solve 3 x2 +5 x2=0
The Napa River flows at a rate of
4km/hr for the
CHAPTER 2
From an Earth-Centered to a Sun-Centered System
CHAPTER OUTLINE
2-1 Science and Its Ways of Knowing
1. It is not easy to define what science is. However, any effort to define it must include its
methods, its historical development, its social co
Informative Self Critique
Please review your recording and hand in a full 2-3 page (no less than 2 pages) typed critical
analysis focusing on your informative speech performance using Informative Speech the
assessment form. I would also like you to assign
Stat 135, Fall 2006
HOMEWORK 1 (due Friday 9/8)
1. Let x1 , x2 , . . . , xn be a list of numbers with mean and SD . Show that 2 = 1 n
n
x2 - 2 i
i=1
2. A class has two sections. Students in Section 1 have an average score of 75 with an SD of 10. Students
Stat 135, Fall 2006
A. Adhikari
HOMEWORK 2 (due Friday 9/15)
In what follows, x.y means Problem y of Chapter x of the text. 1. Consider the situation of sampling without replacement from a population, with the notation used in lecture. Here are some facts
Stat 135, Fall 2006
A. Adhikari
HOMEWORK 3 (due Friday 9/22)
1. Read Section 8.2 and Example A of Section 8.4. There's some redundancy there, but it doesn't hurt to re-read stuff about the Poisson. Then do Problem 8.10. 2. 8.2. You've already done the nec
Stat 135, Fall 2006
A. Adhikari
HOMEWORK 4 (due Friday 9/29)
1. 8.19 2. 8.21a-b. 3. 8.23. 4. 8.27.
Exercises 5 through 10 are one big project broken into pieces. Do them in order, otherwise things won't make sense. And for this HW, turn in your R code for
Stat 135, Fall 2006
A. Adhikari
HOMEWORK 5 (due Friday 10/6)
1. 8.20. Use R to do this one accurately. Be careful about constants when you're working out the distribution of 2 . ^ 2. 8.26. Is your estimate a method of moments estimate? Is it a maximum lik
Stat 135, Fall 2006
A. Adhikari
HOMEWORK 6 (due Friday 10/13)
1. 9.3. Do b accurately using R. Say what the limiting power is as p approaches 0.5; also say what the limiting power is as p approaches 0 and 1. 2. 9.11. Again, use R to do the plots. In each
Stat 135, Fall 2006
A. Adhikari
HOMEWORK 7 (due Friday 10/27)
1. Read the page help(chisq.test) all the way down. There will be stuff you don't understand, but most of it should make sense. 2. 9.34. 3. 9.38. 4. (From Freedman, Pisani, and Purves) In the C
Stat 135, Fall 2006
A. Adhikari
HOMEWORK 8 (due Friday 11/3)
There's nothing strange about doing the Ch 13 problems first. You'll see after you've done them. 1. 13.7. It's easy enough to get R to do the test, so that's not the point. The point is: where's
Stat 135, Fall 2006
HOMEWORK 9 (due Friday 11/17)
You have two weeks to do these problems. If you decide to take a week off and just do them in the second week, you will not find me helpful. On Tuesday 11/14 I will help you with at most three problems, an