Mrs. Waldron
Ch 6.2
BC Calculus
Integration by Substitution
Evaluate the indefinite integral:
1
dx
1.
sec x
2.
( csc
e2u
3. u du
e
4.
1 + t
2
x + 3x 2 ) dx
3
2
Use substitution to evaluate the integral:
5.
6.
2t
7.
x
1 - 4x 2
dx
1 + t 2 dt
cos (7t + 5
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.
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
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
1.6 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
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
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-life situation
2.2 Limits Involving Infinity
North Dakota Sunset
Photo by Vickie Kelly, 2006 Greg Kelly, Hanford High School, Richland, Washington
1 f ( x) = x 1 lim = 0 x x
As the denominator gets larger, the value of the fraction gets smaller. There is a horizontal as
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
2.4 Rates of Change and Tangent Lines
Devils Tower, Wyoming
Photo by Vickie Kelly, 1993 Greg Kelly, Hanford High School, Richland, Washington
The slope of a line is given by:
m=
y x
y
x
The slope at (1,1) can be approximated by the slope of the secant thr
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) f ( a) h
is called the derivative of
f
at
a.
We write:
f ( x ) = lim
h 0
f ( x + h) f ( x) h
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
3.3 Rules for Differentiation
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) = 0
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
Mrs. Waldron
Ch 6.3
BC Calculus
Integration by Parts
Use the formula
u dv = u i v - v du
Evaluate the indefinite integral:
ln x
1. 2 dx
x
2.
ln ( x
2
3.
tan
r dr
4.
3x
x e dx
5.
x
1
+ 1) dx
2
2
cos 2x dx
6. sin ( ln t ) dt
7.
( ln x )
8.
x
9.
x e
10.
Mrs. Waldron
Ch 6.4
Exponential Growth and Decay
BC Calculus
Use separation of variables to solve the initial value problem.
Indicate the domain over which the solution is valid.
1.
dy
= yx and y = 2 when x = 0
dx
2.
dy
= t e y and y = 0 when t = 1
dt
3.
Ms. Waldron
Ch 7.1
Integral as Net Change
Hw: Pg. 386: 5, 7, 9, 10, 19, 23, 29, 31-36
BC Calculus
Activity:
In exercises 1 and 2, the function v(t) is the velocity in m/sec of a particle moving
along the x-axis and starting at the position s(0) = 8. Use a
Mrs. Waldron
Ch 7.2
BC Calculus
Areas in the Plane Worksheet
Find the area of the region enclosed by the two curves f(x) and g(x):
1. f(x) = -3x2 + 5 and g(x) = 2x.
2. f(x) = 4 x2 and g(x) = x2 4
3. f(x) = 2x and g(x) = 2x
4. f(x) = sin x and g(x) = 3 sin
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
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.
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,
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
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
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
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,
8.2 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
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
3.5 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
0
2
0
Now we conne
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