1.3 Exponential Functions
Acadia National Park, Maine
Photo by Vickie Kelly, 2008
Greg Kelly, Hanford High School, Richland, Washington
Although some of todays lecture is from the
book, some of it is not. You must take notes to
be successful in calculus.
3.8 Derivatives of Inverse Trig Functions
Lewis and Clark Caverns, Montana
Photo by Vickie Kelly, 1993
Greg Kelly, Hanford High School, Richland, Washington
2
f ( x) = x x 0
df
= 2x
dx
At x = 2:
2
f ( 2) = 2 = 4
df
( 2) = 2 2 = 4
dx
y
8
6
4
x=y
y= x
( 2,
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 ( )
slope
We could make a graph of the slope:
2
0
Now we connect th
3.4 Velocity, Speed, and Rates of Change
Photo by Vickie Kelly, 2008
Denver & Rio Grande Railroad
Gunnison River, Colorado
Greg Kelly, Hanford High School, Richland, Washington
Consider a graph of displacement (distance traveled) vs. time.
Average velocit
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 differentiable, a fun
3.1
Derivatives
Great Sand Dunes National Monument, Colorado
Photo by Vickie Kelly, 2003
Greg Kelly, Hanford High School, Richland, Washington
lim
h 0
f (a + h) f (a )
h
We write:
is called the derivative of
f ( x ) = lim
h 0
f
at
a.
f ( x + h) f ( x)
h
T
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 motion witho
2.2 Limits Involving Infinity
North Dakota Sunset
Photo by Vickie Kelly, 2006
Greg Kelly, Hanford High School, Richland, Washington
4
1
f ( x) =
x
3
2
1
-4
1
lim = 0
x x
-3
-2
-1
0
1
2
3
4
-1
-2
-3
-4
As the denominator gets larger, the value of the fract
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 reallife situations
2.1
Rates of Change
and Limits
Grand Teton National Park, Wyoming
Photo by Vickie Kelly, 2007
Greg Kelly, Hanford High School, Richland, Washington
Suppose you drive 200 miles, and it takes you 4 hours.
Then your average speed is:
mi
200 mi 4 hr = 50
hr
d
1.6 Trig Functions
Photo by Vickie Kelly, 2008
Black Canyon of the Gunnison
National Park, Colorado
Greg Kelly, Hanford High School, Richland, Washington
Trigonometric functions are used extensively in calculus.
When you use trig functions in calculus, yo
1.5 Functions and Logarithms
Golden Gate Bridge
San Francisco, CA
Photo by Vickie Kelly, 2004
Greg Kelly, Hanford High School, Richland, Washington
A relation is a function if:
for each x there is one and only one y.
A relation is a one-to-one if also:
fo
1.4 Parametric Equations
Mt. Washington Cog Railway, NH
Photo by Greg Kelly, 2005
Greg Kelly, Hanford High School, Richland, Washington
There are times when we need to describe motion (or a
curve) that is not a function.
We can do this by writing equation
Mt. Rushmore, South Dakota
3.9: Derivatives of Exponential and Logarithmic Functions
Photo by Vickie Kelly, 2007
Greg Kelly, Hanford High School, Richland, Washington
Look at the graph of
y=e
x
3
The slope at x=0
appears to be 1.
If we assume this to
be t