10.1
Parametric Functions
6 4 2 -6 -4 -2 0 -2 -4 -6 2 4 6
Greg Kelly, Hanford High School, Richland, Washington
In chapter 1, we talked about parametric equations. Parametric equations can be used to describe motion that is not a function.
x
f
A.P. Calculus BC Formulas 2005-2006
Hanford High School, Richland, Washington revised 3/8/06
1. 2. floor function (def)
x
Greatest integer that is less than or equal to x.
(graph)
3.
ceiling function (def)
Least integer that is greater than or
Hyperbolic Functions
Hyperbolic cosine of x: Hyperbolic sine of x:
cosh x sinh x
e
x
Integrals
e 2
x
x
sinh u du cosh u du sech 2 u du
cosh u C sinh u C tanh u C
csch 2 u du
coth u C sech u C csch u C
ex e 2
sech u tanh u du csch u coth u d
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
z
4 x y
y 1
0 x 2 and 0
We can sketch the graph by putting in the corners where (x=0, y=0
z
100
An Introduction to Partial Derivatives
10
10
y
x
Greg Kelly, Hanford High School, Richland, Washington
z
100
When we have functions with more than one variable, we can find partial derivatives by holding all the variables but one constan
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 f x
x2 sin x
In real life, functions often have more than one independent variable: e
Applications of
Hyperbolic Functions
Greg Kelly, Hanford High School, Richland, Washington
A hanging cable makes a shape called a catenary.
x y b though it looks Even a cosh a like a parabola, it is not
a some constant a) (for parabola!
dy dx
x
10.3 Vector Valued Functions
Greg Kelly, Hanford High School, Richland, Washington
Any vector
v
a, b
can be written as a linear
combination of two standard unit vectors.
i v a, b a, 0 a 1, 0 0, b
1, 0
j
0,1
The vector v is a linear combina
A.P. Calculus:
Hyperbolic Functions
Greg Kelly, Hanford High School, Richland, Washington
Consider the following two functions:
y
e
x
e 2
x
y
e
x
e 2
x
These functions show up frequently enough that they have been given names.
y
e
x
10.6: The Calculus of Polar Curves
r
2sin 2.15 0 16
Try graphing this on the TI-89.
Greg Kelly, Hanford High School, Richland, Washington
To find the slope of a polar curve:
dy dx
dy d dx d
d r sin d d r cos d
r sin r cos
r cos r sin
We us
10.5: 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 directions is to point and say "Go a half mile in t
10.4 Projectile Motion
Fort Pulaski, GA
Photo by Vickie Kelly, 2002 Greg Kelly, Hanford High School, Richland, Washington
One early use of calculus was to study projectile motion.
In this section we assume ideal projectile motion: Constant force o
10.2
Vectors in a 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 that we measure that have magnitude but not direction ar