2.1 day 2: Step Functions
"Miraculous Staircase" Loretto Chapel, Santa Fe, NM
Two 360o turns without support!
Photo by Vickie Kelly, 2003 Greg Kelly, Hanford High School, Richland, Washington
"Step functions" are sometimes used to describe real-l
4.1 Extreme Values of Functions
Greg Kelly, Hanford High School Richland, Washington
The textbook gives the following example at the start of chapter 4: The mileage of a certain car can be approximated by:
m v
0.00015v3 0.032v 2 1.8v 1.7
At what
3.6
The Chain Rule
Greg Kelly, Hanford High School, Richland, Washington
Photo by Vickie Kelly, 2002
U.S.S. Alabama Mobile, Alabama
Photo by Vickie Kelly, 2002
Greg Kelly, Hanford High School, Richland, Washington
We now have a pretty good list
3.8 Derivatives of Inverse Trig Functions
Lewis and Clark Caverns, Montana
Photo by Vickie Kelly, 1993 Greg Kelly, Hanford High School, Richland, Washington
f x x2 x df 2x dx
At x = 2:
0
y
8
y
x2
6
4
2, 4
m 4
y x
f 2 df 2 dx
22
4
2 0 0
Mt. Rushmore, South Dakota
3.9: Derivatives of Exponential and Logarithmic Functions
Photo by Vickie Kelly, 2001 Greg Kelly, Hanford High School, Richland, Washington
Look at the graph of
y
e
3
x
The slope at x=0 appears to be 1.
2
If we assu
3.3 Differentiation Rules
Colorado National Monument
Photo by Vickie Kelly, 2003 Greg Kelly, Hanford High School, Richland, Washington
If the derivative of a function is its slope, then for a constant function, the derivative must be zero.
d c dx
2.4 Rates of Change and Tangent Lines
Devil's Tower, Wyoming
Photo by Vickie Kelly, 1993 Greg Kelly, Hanford High School, Richland, Washington
The slope of a line is given by:
16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 1 2 3 4
m
y x
y
x
The slope
3.2 Differentiability
Arches National Park
Photo by Vickie Kelly, 2003 Greg Kelly, Hanford High School, Richland, Washington
Arches National Park
Photo by Vickie Kelly, 2003 Greg Kelly, Hanford High School, Richland, Washington
To be differentiabl
3.1
Photo by Vickie Kelly, 2003
Derivatives
Greg Kelly, Hanford High School, Richland, Washington
Great Sand Dunes National Monument, Colorado
lim
h 0
f a h h
f a
is called the derivative of
f at a .
We write:
f x
lim
h 0
f a h h
f a
"Th
4.3 Using Derivatives for Curve Sketching
Old Faithful Geyser, Yellowstone National Park
Photo by Vickie Kelly, 1995 Greg Kelly, Hanford High School, Richland, Washington
4.3 Using Derivatives for Curve Sketching
Yellowstone Falls, Yellowstone Nat
4.2
Mean Value Theorem for Derivatives
Teddy Roosevelt National Park, North Dakota
Photo by Vickie Kelly, 2002 Greg Kelly, Hanford High School, Richland, Washington
Mean Value Theorem for Derivatives If f (x) is a differentiable function over [a,b
4.5: Linear Approximations, Differentials and Newton's Method
For any function f (x), the tangent is a close approximation of the function for some small distance from the tangent point.
y
f x
f a
We call the equation of the tangent the lineariza
3.4 Velocity, Speed, and Rates of Change
Greg Kelly, Hanford High School, Richland, Washington
Consider a graph of displacement (distance traveled) vs. time.
Average velocity can be found by taking:
distance (miles)
B A
s t
change in position c
3.5 Derivatives of Trig Functions
London Bridge, Lake Havasu City, Arizona
Photo by Vickie Kelly, 2001 Greg Kelly, Hanford High School, Richland, Washington
Consider the function
y sin
We could make a graph of the slope:
slope
1
2
0
1
4.4 Optimization
Buffalo Bill's Ranch, North Platte, Nebraska
Photo by Vickie Kelly, 1999 Greg Kelly, Hanford High School, Richland, Washington
A Classic Problem You have 40 feet of fence to enclose a rectangular garden along the side of a barn. Wha
3.7 Implicit Differentiation
Niagara Falls, NY & Canada
Photo by Vickie Kelly, 2003 Greg Kelly, Hanford High School, Richland, Washington
Before we start, we are going to load the Calculus Tools flash application software to your calculator. 1. Conn
2.2 Limits Involving Infinity
Greg Kelly, Hanford High School, Richland, Washington
4
f x 1 lim x x
1 x
-4 -3 -2 -1
3 2 1 0 -1 -2 -3 -4 1 2 3 4
0
As the denominator gets larger, the value of the fraction gets smaller. There is a horizontal asy
2.1: Rates of Change & Limits
Greg Kelly, Hanford High School, Richland, Washington
Suppose you drive 200 miles, and it takes you 4 hours.
Then your average speed is:
200 mi
4 hr
x t
mi 50 hr
average speed
distance elapsed time
If you look at
2.3 Continuity
Grand Canyon, Arizona
Photo by Vickie Kelly, 2002 Greg Kelly, Hanford High School, Richland, Washington
Most of the techniques of calculus require that functions be continuous. A function is continuous if you can draw it in one motio