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
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, )
( , )
8.4 day one
Improper Integrals
Greg Kelly, Hanford High School, Richland, Washington
Until now we have been finding integrals of continuous functions over closed intervals.
Sometimes we can find integrals for functions where the function or the limits are
8.3 Relative Rates of Growth
Greg Kelly, Hanford High School, Richland, Washington
y = e x grows very fast. The function
We could graph it on the chalkboard: If x is 3 inches, y is about 20 inches:
( 3, 20 )
Wte64 inches, the y-value We have gone less tha
8.2 Day 2: Identifying Indeterminate Forms
Photo by Vickie Kelly, 2008
Brooklyn Bridge, New York City
Greg Kelly, Hanford High School, Richland, Washington
What makes an expression indeterminate?
Consider:
We can hold one part of the expression constant
8.2 day 1 LHpitals Rule
Actually, LHpitals Rule was developed by his teacher Johann Bernoulli. De lHpital paid Bernoulli for private lessons, and then published the first Calculus book based on those lessons.
Guillaume De l'Hpital 1661 - 1704
Greg Kelly,
8.1: Sequences
Craters of the Moon National Park, Idaho
Photo by Vickie Kelly, 2008 Greg Kelly, Hanford High School, Richland, Washington
A sequence is a list of numbers written in an explicit order.
cfw_ an = cfw_ a1, a2 , a3, . , an , .
nth term
Any r
Chapter 7 Extra Topics
Crater Lake, Oregon
Photo by Vickie Kelly, 1998 Greg Kelly, Hanford High School, Richland, Washington
Centers of Mass:
Fg
d
Take Superior, Washburn, WI L orque is a function of force and distance. Photo by Vickie Kelly, 2004 (Torque
7.5 Fluid Pressure and Forces
Georgia Aquarium, Atlanta
Photo by Vickie Kelly, 2006 Greg Kelly, Hanford High School, Richland, Washington
What is the force on the bottom of the aquarium? 2 ft
Force = weight of water = density volume
lb = 62.5 3 2 ft 3 ft
Hoover Dam Nevada & Arizona
Photo by Vickie Kelly, 2004
7.5 part 1 Work and Pumping Liquids
Greg Kelly, Hanford High School, Richland, Washington
Hoover Dam Powerhouse Nevada & Arizona
Photo by Vickie Kelly, 2004 Greg Kelly, Hanford High School, Richland,
A.P Calculus Worksheet: Areas of Surfaces of Revolution
Find the areas of the surfaces generated by revolving the curves in problems 1-4 about the axes indicated: 1. y = x / 2, 0 x 4 , about the x -axis. Check your result with a formula from geometry. 2.
7.4 Day 2
Surface Area
(Photo not taken by Vickie Kelly)
Greg Kelly, Hanford High School, Richland, Washington
Surface Area:
ds
r
Consider a curve rotated about the x-axis: The surface area of this band is:
2 r ds
The radius is the y-value of the function
7.4 Day 1 Lengths of Curves
Golden Spike National Historic Site, Promontory, Utah
Photo by Vickie Kelly, 1999 Greg Kelly, Hanford High School, Richland, Washington
Lengths of Curves:
ds dx
dy
If we want to approximate the length of a curve, over a short d
7.3 day 3 The Shell Method
Grows to over 12 feet wide Japanese Spider Crab and Aquarium, Atlanta Georgialives 100 years.
Photo by Vickie Kelly, 2006 Greg Kelly, Hanford High School, Richland, Washington
y = x2 + 1
Find the volumeof dy + 4 4 ( y 1) the reg
7.3 day 2 Disk and Washer Methods
Limerick Nuclear Generating Station, Pottstown, Pennsylvania
Photo by Vickie Kelly, 2003 Greg Kelly, Hanford High School, Richland, Washington
y= x
Suppose I start with this curve. My boss at the ACME Rocket Company has a
7.3 Day One: Volumes by Slicing 7.3
Little Rock Central High School, Little Rock, Arkansas
Photo by Vickie Kelly, 2001 Greg Kelly, Hanford High School, Richland, Washington
3
Find the volume of the pyramid: Consider a horizontal slice through the pyramid.
7.2 Areas in the Plane
Gateway Arch, St. Louis, Missouri
Photo by Vickie Kelly, 2003 Greg Kelly, Hanford High School, Richland, Washington
y1 = 2 x
2
How can we find the area between these two curves?
y2 = x
We could split the area into several sections,
Photo by Vickie Kelly, 2006
Greg Kelly, Hanford High School, Richland, Washington
A honey bee makes several trips from the hive to a flower garden. The velocity graph is shown below. What is the total distance traveled by the bee?
200 + 200 + 200 + 100 =
6.5 day 2
Logistic Growth
Columbian Ground Squirrel Glacier National Park, Montana
Photo by Vickie Kelly, 2004 Greg Kelly, Hanford High School, Richland, Washington
y = y0 e kt We have used the exponential growth equation
to represent population growth.
T
6.5 day 1 Partial Fractions
The Empire Builder, 1957
Greg Kelly, Hanford High School, Richland, Washington
1
5x 3 x 2 2 x 3 dx
This would be a lot easier if we could re-write it as two separate terms.
5x 3 A B = + ( x 3) ( x + 1) x 3 x + 1
These are calle
Glacier National Park, Montana
Photo by Vickie Kelly, 2004
6.4 Exponential Growth and Decay
Greg Kelly, Hanford High School, Richland, Washington
The number of bighorn sheep in a population increases at a rate that is proportional to the number of sheep p
6.3 Integration By Parts
Badlands, South Dakota
Photo by Vickie Kelly, 1993
Greg Kelly, Hanford High School, Richland, Washington
6.3 Integration By Parts
Start with the product rule:
d dv du ( uv ) = u + v dx dx dx
d ( uv ) = u dv + v du d ( uv ) v du =
6.2 Integration by Substitution & Separable Differential Equations
M.L.King Jr. Birthplace, Atlanta, GA
Photo by Vickie Kelly, 2002
Greg Kelly Hanford High School Richland, Washington
The chain rule allows us to differentiate a wide variety of functions,
6.1 day 2 Eulers Method
Leonhard Euler made a huge number of contributions to mathematics, almost half after he was totally blind. (When this portrait was made he had already lost most of the sight in his right eye.) Leonhard Euler 1707 - 1783
Greg Kelly,
6.1 day 1: Antiderivatives and Slope Fields
Greg Kelly, Hanford High School, Richland, Washington
First, a little review: Consider: then:
y = x2 + 3
y = 2 x
or
y = x2 5
y = 2 x
It doesnt matter whether the constant was 3 or -5, since when we take the deri
5.5 Numerical Integration
Mt. Shasta, California
Photo by Vickie Kelly, 1998 Greg Kelly, Hanford High School, Richland, Washington
Using integrals to find area works extremely well as long as we can find the antiderivative of the function. Sometimes, the
5.4 Fundamental Theorem of Calculus
Morro Rock, California
Photo by Vickie Kelly, 1998 Greg Kelly, Hanford High School, Richland, Washington
Here is my favorite calculus textbook quote of all time, from CALCULUS by Ross L. Finney and George B. Thomas, Jr.
5.3 Definite Integrals and Antiderivatives
Greg Kelly, Hanford High School, Richland, Washington
Page 269 gives rules for working with integrals, the most important of which are: 1.
a a
b
a
f ( x ) dx = f ( x ) dx
b
a
Reversing the limits changes the si
5.2 Definite Integrals
Greg Kelly, Hanford High School, Richland, Washington
1 V = t2 +1 8
When we find the area under a curve by adding rectangles, the answer is called a Rieman sum. The width of a rectangle is called a subinterval.
subinterval partition
5.1 Estimating with Finite Sums
Greenfield Village, Michigan
Photo by Vickie Kelly, 2002 Greg Kelly, Hanford High School, Richland, Washington
Consider an object moving at a constant rate of 3 ft/sec. Since rate . time = distance:
3t = d
If we draw a grap