Long answer questions
For each long answer problem, you must give a full explanation and justification of your
answer to receive credit. A list of computations is not sufficient to gain credit!
x 10
. Find f 0 (x) and f 00 (x). Then find all the points wh
Warmup
Area between curves
1. Calculate the area between the x axis and the curve
y = x 2 + 5x 6 between x = 1 and x = 2.
(Your answer should be positive we want area.)
2. Calculate the area of the region enclosed between the curve
y = x 2 + 5x 6 and th
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
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
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)
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?),
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.
Math 3 Fall 2014
Final Exam
November 21, 2014
Your name (please print):
Circle your section:
Pauls
Cai
Andrews
Hein
Instructions: This is a closed book, closed notes exam. Use of calculators is not permitted.
Except for the multiple choice questions, you
Math 3 Fall 2014
Midterm 2
November 4, 2014
Your name (please print):
Circle your section:
Pauls
Cai
Andrews
Hein
Instructions: This is a closed book, closed notes exam. Use of calculators is not permitted.
Except for the multiple choice questions, you mu
Math 3 Fall 2014
Midterm 1
October 14, 2014
Your name (please print):
Circle your section:
Pauls
Cai
Andrews
Hein
Instructions: This is a closed book, closed notes exam. Use of calculators is not
permitted. Except for the multiple choice questions, you mu
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
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