3
POLYNOMIAL AND RATIONAL FUNCTIONS
52
In general, let r(x) be a rational function
1. The vertical asymptotes of r(x) are the roots of the denominator.
2. The horizontal asymptotes are determined as follows:
If the degree of the top is larger than the de
Exponential growth and decay
Warmup
1. If (A + B )x
2A = 3x + 1 for all x , what are A and B ?
(Hint: if its true for all x , then the coe cients have to match up,
i.e. A + B = 3 and
2A = 1.)
2. Find numbers (maybe not integers) A and B which satisfy
3x
Modeling with Dierential Equations:
Introduction to the Issues
Warmup
Do you know a function. . .
. . . whose rst derivative is the same as the function itself, i.e.
d
f (x ) = f (x )?
dx
. . . whose rst derivative is negative of the function, i.e.
d
f (
Antiderivatives and Initial Value Problems
Warm up
If
d
f (x ) = 2x , what is f (x )?
dx
Can you think of another function that f (x ) could be?
If
d
f (x ) = 3x 2 + 1, what is f (x )?
dx
Can you think of another function that f (x ) could be?
Denition
An
Op#miza#on
Warm up
Sketch the graph of
f (x ) = ( x
3)(x
2)(x
1) = x 3
6x 2 + 11x
6
over the interval [1, 4]. Mark any critical points and inection points.
What is the absolute maximum over this interval? What is the absolute
minimum overp interval?
this
Math 3  Day 1  WARMUP
Calculate the given functions at the given xvalues, and then plot the corresponding points.
y
f ( x) = x2
x f ( x)
2
1
0
1
x
2
y
f (x) = (x + 1)
x
f ( x)
2
2
1
0
1
x
f ( x) = x2 + 1
x
f ( x)
y
2
1
0
1
2
x
Welcome to Math 3!
http:/
Going between graphs of
functions and their derivatives:
Mean value theorem, Rolles theorem, and
intervals of increase and decrease
Recall: The Intermediate Value Theorem
Suppose f is continuous on a closed interval [a, b ].
If
f (a ) < C < f (b )
or
f (a
Newtons Method and Linear Approximations
Newtons Method
Step 1: Pick a place to start. Call it x0 .
Step 2: The tangent line at x0 is y = f (x0 ) + f 0 (x0 ) (x
where this intersects the x axis, solve
0 = f ( x0 ) + f 0 ( x0 ) ( x
x0 )
to get
x = x0
x0 )
Trigonometric functions
Step one: similar triangles
Two similar triangles have the same set
of angles, and have the properties that
c
a
b
A
a
=,
B
b
B
b
A
a
= , and
=.
C
c
C
c
Dene
C
B
A
cos() =
b
c
and
sin() =
Then let
tan() =
sin()
a
=,
cos()
b
csc() =
Exponential and Logarithmic Functions
The Basics
If n and m are positive integers.
an = a a a

cfw_z
(WeBWoRK: a^ n or a n)
n
Some identities:
a n a m = a n +m
n
n
( a n ) m = a n m
n
(Notice: am means a(m ) , since (am ) can be written another way)
Related Rates
Example
Suppose you has a 5m ladder resting against a wall.
y
5m
x
dx
dt
dy
dt
x2
Move the base out at 2 m/s:
How fast does the top move down the wall?
To solve, we need to relate the variables:
=2
=?
+ y 2 = 52
0x 5
Problem:
If x 2 + y 2 =
Curve Sketching
Warm up
Below are pictured six functions: f , f 0 , f 00 , g , g 0 , and g 00 . Pick out
the two functions that could be f and g , and match them to their
rst and second derivatives, respectively.
(a)
(b)
(c)
3
3
3
2
2
2
1
1
1
3
3
2
1
We take a lot of our notation from Leibniz.
independently coinvented calculus, taking a slightly
dierent point of view (innitesimal calculus) but
also studied rates of change in a general setting.
Gottfried Wilhelm Leibniz (16461716)
Galileo (15641642)
Math 3, Midterm 1 Solutions
October 21, 2009
For the multiple choice questions, we omit the choices and just calculate the answer.
1. For the function
x1
f (x) =
= x1 x2 ,
x2
what is its domain?
Solution. This function is dened everywhere except x = 0; th
Slope Fields and Eulers Method
Warm up
Suppose dy = y
dx
following points:
x
2
2
2
1
1
0
0
0
1
y
0
1
1
1
1
2
0
2
1
x
dy
dx
1. Sketch part of the slope eld for the
Warm up
Suppose dy = y
dx
following points:
x
2
2
2
1
1
0
0
0
1
y
0
1
1
1
1
Arc$Length
Suppose you want to know what the length of a curve y = f (x ) is
from the point (a, f (a) to the point (b , f (b ):
y=f(x)
l
one
piece
!
x
x2
x1
x0
.
xn
y
xi
` = lim
n!1
( `) i =
i =1
d` =
xi+1
Let n go to 1
Slice!
n
X
x
Z
x =b
d`
dl
x =a
dy
p
3
POLYNOMIAL AND RATIONAL FUNCTIONS
To sketch the rest we can do a table of values. What we end up with is
This graph never crosses either of the axes but gets close to both of them.
49
3
POLYNOMIAL AND RATIONAL FUNCTIONS
Transformations of
50
1
x
ax + b
3
POLYNOMIAL AND RATIONAL FUNCTIONS
43
Thus, x3 + 3x2 + 3x + 1 = (x + 1)(x2 + 2x + 1) = (x + 1)(x + 1)2 = (x + 1)3 . So f (x) has
only one root, x = 1.
Here, you were given one of the roots. Sometimes you have to nd one yourself. Here is a
good general ru
4
Exponential and Logarithmic Functions
4.1
Exponential Functions
Denition 4.1 If a > 0 and a 6= 1, then the exponential function with base a is given
by f (x) = ax .
x
1
Examples: f (x) = 2 , g(x) = 10 , h(x) =
.
3
x
x
Graphs of Exponential Functions
x
4
EXPONENTIAL AND LOGARITHMIC FUNCTIONS
58
The most common exponentials used in mathematics are
1. Base 2: f (x) = 2x
2. Base 10: f (x) = 10x
3. Base e: f (x) = ex
The number e is the value that the expression
1
1+
n
n
approaches as n ! 1.
e 2.7182818.
Ap
3
POLYNOMIAL AND RATIONAL FUNCTIONS
46
What luck! A hit on the rst try. Thus we should divide the polynomial by x
1
0
1
6
1
1
6
1
1
7
6
1:
0
Note that we are relieved to get 0 as a remainder  that conrms that 1 is a root and that
we didnt make a mistake
3
3.1
Polynomial and Rational Functions
Polynomial Functions and their Graphs
So far, we have learned how to graph polynomials of degree 0, 1, and 2. Degree 0 polynomial
functions are things like f (x) = 2, which is a straight horizontal line with constan
The FUNDAMENTAL Theorem of Calculus
(yay!)
Warmup
Suppose a particle is traveling at velocity v (t ) = t 2 from t = 1 to
t = 2. if the particle starts at y (0) = y0 ,
1. what is the function y (t ) which gives the particles position as
a function of time
The Denite Integral
The Area Problem
Upper and Lower Sums
Suppose we want to use rectangles to approximate the area under
the graph of y = x + 1 on the interval [0, 1].
2
2
1.5
1.5
1
1
0.5
0.5
0 0.2
0.4
0.6
0.8
1
0 0.2
Upper Riemann Sum
31/20
>
0.4
0.6
0.
Modelling Accumulations
The purpose of calculus is twofold:
1. to nd how something is changing, given what its doing;
2. to nd what something is doing, given how its changing.
We did derivatives
(a) algebraically (derivative rules, what is the function?),