R E F E R E N C E PA G E S
Cut here and keep for reference
ALGEBRA
GEOMETRY
Arithmetic Operations
Geometric Formulas
a
c
ad " bc
" !
b
d
bd
a
a
b
d
ad
! ) !
c
b
c
bc
d
a!b " c" ! ab " ac
a"c
a
c
! "
b
b
b
Formulas for area A, circumference C, and volume V
The Ratio Test
The Ratio Test is a test that can be used to determine if an infinite series converges.
We can apply it to series where the terms are numbers; we can also apply it to power
series.
To understand the Ratio Test, we look back to geometric s
Section 8.5: Convergence of Power Series
Now we apply the Ratio Test to power series power series. We will only look at power
series centered at x = 0, that is, power series of the form
X
cn x n .
n=0
The Ratio Test will help us to answer this extremely
Brief Introduction to Power Series and Taylor Series
Weve seen how to use Taylor polynomials to approximate a function. It appears that,
in such cases, the accuracy of the approximation improves the degree of the polynomial
increases, at least for the va
Excel Quiz
2004. 6. 2
1. Given below are seven observations collected in a regression study on two variables, x
(independent variable) and y (dependent variable). Use Excels Regression Tool to
answer the following questions.
a.
x
y
2
12
3
9
6
8
7
7
8
6
7
Introduction to Taylor Polynomials
I. Taylor Polynomials of Degree 1
Let f (x) be the function were trying to approximate. Suppose f is dierentiable at
x = a. Lets start by looking at the tangent line to the graph of f at x = a. It has
slope f 0 (a) and
Returning to Taylor Series (Section 8.7)
We end the course by returning to Taylor series. Recall the definition of the Taylor
series of a function f (x) centered at x = a:
Suppose that f (x) has derivatives of all orders at x = a. Then the Taylor
series
Math 10b
Solutions to Self-Quiz on Taylor Series (Section 8.7)
1
= (1 + x)1 . Note that f (0) = 1. Compute the first few derivatives of f (x)
1+x
and evaluating them at x = 0:
1. f (x) =
f 0 (x) = (1 + x)2 , so f 0 (1) = 1
f 00 (x) = 2(1 + x)3 , so f 00
Solutions to Self-Quiz on Section 8.1
n
x
. Look at the related function f (x) =
. Since
n 1 2n
1 2x
x
1
lim
=
,
x 1 2x
2
1
the the sequence converges to .
2
1. (a) lim
(b) lim 3 + (1)n . The sequence looks like cfw_ 2, 4, 2, 4, 2, . . . , so it diverges.
Math 10b
Solutions to Self-Quiz on Taylor Polynomials
1. f (x) = e2x . Note that f (0) = 1. Compute and evaluate the appropriate derivatives:
f 0 (x) = 2e2x f 0 (0) = 2
f 00 (x) = 4e2x f 00 (0) = 4
f 000 (x) = 8e2x f 000 (0) = 8
f (4) (x) = 16e2x f (4
Solutions to Self-Quiz on Section 8.2
1. (a)
(b)
(c)
(d)
X
1 n
1
. This series is geometric with a = 3 and r = . Since |r| < 1,
5
5
n=0
3
3
5
the series converges to
= 6 = .
1
1 ( 5 )
2
5
3
X
1 3 n
1
3
. This series is geometric with a =
and r = . Sinc
Solutions to Self-Quiz on the Ratio Test and Convergence of Power Series (Sections 8.4 and 8.5)
1. (a)
X
n=0
(1)n 9n
. Use the ratio test:
n!
a
n+1
lim
n a
n
=
lim
n
(1)n+1 9n+1
(n + 1)!
(1)n 9n
n!
9n+1
(n + 1)!
= lim
= lim
n
n
9n
n!
= lim
n
9n+1
n!
CHAPTER 7SAMPLING AND SAMPLING DISTRIBUTIONS
MULTIPLE CHOICE
1.
From a group of 12 students, we want to select a random sample of 4 students to serve
on a university committee. How many different random samples of 4 students can be selected?
a.
48
b.
20,7
Using a Taylor Polynomial to Approximate the Solution to a Differential Equation
Example. Find an 8th degree Taylor polynomial that approximates the solution to the
differential equation f 00 (x) + f (x) = 0, subject to the following initial conditions: f