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
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
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
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 lif
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 b
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
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 (
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 =
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
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 th
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
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 t
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 tri
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 i
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
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 of
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 (
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 inte
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
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 Har
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 p