8.4 day 2
Tests for Convergence
Riverfront Park, Spokane, WA
Greg Kelly, Hanford High School, Richland, Washington Photo by Vickie Kelly, 2006
Review:
1
dx P x
P>0
(P is a constant.)
b 1 lim b P + 1 P +1
P +1
P +1
1
x
P
dx
b P +1 gets bigger If P 1 then
Hyperbolic Functions
Scottys Castle, Death Valley, CA
Photo by Vickie Kelly, 2005 Greg Kelly, Hanford High School, Richland, Washington
Consider the following two functions:
e e y= 2
x
x
e +e y= 2
x
x
These functions show up frequently enough that they ha
Applications of Hyperbolic Functions
Greg Kelly, Hanford High School, Richland, Washington
A hanging cable makes a shape called a catenary.
x y ven + a cosh E = b though it looks a like a parabola, it is not
a some constant a) (for parabola!
dy x = sinh d
Greg Kelly, Hanford High School, Richland, Washington
The functions that we have studied this year have all
been functions of one independent variable:
eg:
f ( x ) = x2
f ( x ) = sin ( x )
In real life, functions often have more than one
independent varia
z
100
10
An Introduction to Partial Derivatives
y
x
10
Greg Kelly, Hanford High School, Richland, Washington
z
100
When we have functions with more than one variable, we can find partial derivatives by holding all the variables but one constant.
f ( x, y
Double Integration
Greg Kelly, Hanford High School, Richland, Washington
f ( x, y ) = 4 x y
Find the volume under this surface between 0<x<2 and 0<y<1.
f ( x, y ) = 4 x y
z
0 < x < 2 and 0 < y < 1
We can sketch the graph by putting in the corners where (x
A.P. Calculus Formulas 2008-2009
Hanford High School, Richland, Washington revised 8/25/08
1. 2. floor function (def) x (graph) Greatest integer that is less than or equal to x.
3.
ceiling function (def) Least integer that is greater than or equal to x.
Hyperbolic Functions
Hyperbolic cosine of x: Hyperbolic sine of x: cosh x =
sinh x =
e +e 2
x x
Integrals
e x e x 2
sinh u du = cosh u + C cosh u du = sinh u + C sech u du = tanh u + C
2
csch u du = coth u + C sech u tanh u du = sech u + C csch u coth u
HW 1.2.1: Function Foundations (Domain and Range)
Inequality
5 h 10
Set Builder Notation
h | 5 h 10
Interval notation
(5, 10]
5 h 10
h | 5 h 10
h | 5 h 10
h | h 10
h | h 10
h| h
[5, 10)
5 h 10
h 10
h 10
all real numbers
(5, 10)
( ,10)
[10, )
( , )
10.3 day 2
Calculus of
Polar Curves
Lady Bird Johnson Grove,
Redwood National Park, California
Photo by Vickie Kelly, 2007
Greg Kelly, Hanford High School, Richland, Washington
Try graphing this
on the TI-89.
r = 2sin ( 2.15 )
0 16
To find the slope of a
10.3 day 1
Polar Coordinates
Greg Kelly, Hanford High School, Richland, Washington
One way to give someone directions is to tell them to
go three blocks East and five blocks South.
Another way to give directions is to point and say Go a
half mile in that
10.2 day 1: Vectors in the Plane
Mesa Verde National Park, Colorado
Photo by Vickie Kelly, 2003 Greg Kelly, Hanford High School, Richland, Washington
Warning: Only some of this is review.
Quantities that we measure that have magnitude but not direction ar
Ch.8 Extra Topic: Trigonometric Substitutions
Monticello (Thomas Jeffersons home), Alexandria, VA
Photo by Vickie Kelly, 2004 Greg Kelly, Hanford High School, Richland, Washington
We can use right triangles and the pythagorean theorem to simplify some pro
9.1 9.1 Power Series
Photo by Vickie Kelly, 2003
Greg Kelly, Hanford High School, Richland, Washington
Start with a square one Start unit by one unit: unit
111111 + + + + + + = 1 2 4 8 16 32 64
This is an example of an infinite series. infinite
1 2
1 8 1
9.2: Taylor Series
Brook Taylor was an accomplished musician and painter. He did research in a variety of areas, but is most famous for his development of ideas regarding infinite series.
Brook Taylor 1685 - 1731
Greg Kelly, Hanford High School, Richland,
9.2 day 2 Finding Common Maclaurin Series
Liberty Bell, Philadelphia, PA
Photo by Vickie Kelly, 2003 Greg Kelly, Hanford High School, Richland, Washington
There are some Maclaurin series that occur often enough that they should be memorized. They are on y
9.3 Taylors Theorem: Error Analysis for Series
Tacoma Narrows Bridge: November 7, 1940
Greg Kelly, Hanford High School, Richland, Washington
Taylor series are used to estimate the value of functions (at least theoretically - now days we can usually use th
9.4
Radius of Convergence
Photo by Vickie Kelly, 2008
Abraham Lincolns Home Springfield, Illinois
Greg Kelly, Hanford High School, Richland, Washington
Convergence The series that are of the most interest to us are those that converge. Today we will consi
9.5
Testing Convergence at Endpoints
Petrified Forest National Park, Arizona
Photo by Vickie Kelly, 2007 Greg Kelly, Hanford High School, Richland, Washington
Remember:
The Ratio Test:
an +1 =L If an is a series with positive terms and lim n a n
then: The
10.1 Parametric functions
Photo by Vickie Kelly, 2008
Mark Twains Boyhood Home Hannibal, Missouri
Greg Kelly, Hanford High School, Richland, Washington
Photo by Vickie Kelly, 2008
Mark Twains Home Hartford, Connecticut
Greg Kelly, Hanford High School, Ric
Precalculus
Wkst- Linear Functions and Linear Modeling
Name:_
Date:_
1. A town's population has been growing linearly. In 2003, the population was 45,000, and the population has
been growing by 1700 people each year.
a. Write an equation, P t , for the po