Question 6. (See Week 2 Lecture page Transformations for a starting point.) Graph the
function y = -2|x 7| + 4 as follows: a. Starting with the expression for the absolute value
function y = |x| deduce the sequence of transformations needed to create the
Summary: Inverse Functions
One-to-One Function
Inverse Functions
Notation
Definition of the Inverse
of a Function
Finding the Inverse
Function for the One-to-
One Function f(x)
Graphs of f(x) and f1(x)
Deciding Whether a
Function is One-to-One
Theorem
S
Section 4.3: Complex Zeros of Polynomials Problem Set
Nothing will work unless you do. - Maya Angelou
Definition of i The imaginary number i is defined as i = J:
Powers of i
Definition of E for b > o If b is a positive real number, then JT =iJl;
> Pro
Section 3.1: Definition of a Function; Evaluating Functions Problem Set
The way we see the problem is the problem. - $.R. Covey
Relation A relation is any set of ordered pairs (x, y).
. The set of first elements in the ordered pairs is the d
Section 4.4: Graphing Rational Functions and Solving Rational Inequalities Problem Set
Great spirits have always encountered violent opposition from mediocre minds. - Albert Einstein
JL.
Graphs of Rational
Functions
1
_ . _; .
yx,l (n is odd) y_x" (nis
Section 3.2: Graphing Functions Problem Set
Life is change. Growth is optional. Choose wisely. -Karen Kaiser Clark
Finding the Domain Draw vertical lines from the curve to the x-axis.
from a Graph The set of points where the vertical lines intersect the
Section 3.1: The Domain of a Function Problem Set
We are what we repeatedly do. Excellence, then, is not an act, but a habit. - Aristotle
Finding the Domain of a Function Algebraically. The domain of a function depends upon the type of the function, as in
MTH 105 Exam 2
Name _
Directions: Show all work for full credit. Books and notes are not allowed in this exam. You may use a
calculator, but if a problem asks for an exact answer, you may not use a calculator. If you are unclear
about anything, ask.
Turn
MTH 105 Exam 3
Name _
Directions: Show all work for full credit. Books and notes are not allowed in this exam. You may use a
calculator, but if a problem asks for an exact answer, you may not use a calculator. If you are unclear
about anything, ask. Turn
MTH 105 Exam 4
Name _
Directions: Show all work for full credit. Books and notes are not allowed in this exam. You may use a
calculator, but if a problem asks for an exact answer, you may not use a calculator. If you are unclear
about anything, ask. Turn
MTH 105 Exam 1
Name _
Directions: Show all work for full credit. Books and notes are not allowed in this exam. You may use a
calculator, but if a problem asks for an exact answer, you may not use a calculator. If you are unclear
about anything, ask.
Turn
Application for Employment
VALET
IN ORDER TO PROCESS YOUR VALET APPLICATION, WE NEED THE FOLLOWING:
1. _ Valet Parker Uniform Deduction Form
2. _ Application Completed and Signed
3. _ Back Page of Employee Handbook - Signed
MAT 1730
4.5 Limits at Infinity
def: Let L be a real number.
1. lim f ( x) L means that for each 0 , there exists M 0 such that f ( x) L whenever x M .
x
2. lim f ( x) L means that for each 0 , there exists N 0 such that f ( x) L whenever x N .
x
3. lim
MAT 1730
4.6 Curve Sketching
def: The end behavior of a graph is what the function does as x approaches .
The graph could approach a constant, another function, or just shoot up/down.
Note: Rational expressions approach either a horizontal asymptote or a
MAT 1730
5.3 Riemann Sums & Definite Integrals
def: Let f be defined on [a, b] , and let be a partition [a, b] , (not necessarily of equal lengths), given by
a x0 x1 x2 xn1 xn b , where xi is the width of the ith subinterval [ xi 1 , xi ] .
The Riemann su
MAT 1730
4.3 The First Derivative Test
def: A function f is increasing on an interval if for any numbers a, b in the interval, a b implies
f (a) f (b) .
def: A function f is decreasing on an interval if for any numbers a, b in the interval, a b implies
f
MAT 1730
2.2: Limits: Definition, Graphically, Numerically
def: Let f be a function defined on an open interval containing c, except possibly at c, and let L be a real
number.
lim f ( x) L means that for each 0 , there exists 0 such that f ( x) L when 0 x
MAT 1730
5.1 Antiderivatives/Indefinite Integration
Write a function F ( x) whose derivative is f ( x) 2 x .
Write a function F ( x) whose derivative is f ( x) sin x .
def: A function F in an antiderivative of f on an interval I if F ( x) f ( x) for all x
MAT 1730
3.3: Product Rule, Quotient Rule, & Higher Derivatives.
e.g. Find the derivative of f ( x) x 2 .
e.g. Find the derivative of g ( x) 2 x 3
e.g. Find the derivative of h( x) x 2 (2 x 3)
Lets derive a formula for products:
d
f ( x) g ( x) lim
dx
1
MAT 1730
Recall:
5.7: Natural Logarithm Integration
1
u du
Note: You may need to:
1. Make a substitution, for the u or an expression of u.
2. Simplify the integrand (vast algebra/trig skills).
e.g. Find or evaluate the integral:
1
a.
dx
3x 2
c.
2 x2 5x
MAT 1730
5.6: Numerical Integration
You can approximate definite integrals on your calculator.
On the TI 83/84:
b
a
f ( x)dx fnInt ( f ( x), x, a, b)
fnInt is found in the MATH menu.
Use X , T , , n to enter your variable.
e.g. Use your calculator to appr
MAT 1730
5.5 Integration by Substitution
Recall: The Chain Rule states that
d
F (u )
dx
e.g. We know that
d 7x
e =
dx
So, what is 7e7 x dx ?
e.g. We know that
d
( x 2 5)3
dx
What is 6 x( x 2 5)2 dx ?
Theorem: Let u be a function of x, whose range is a
MAT 1730
2.3: Limits: Analytically
e.g. Evaluate the limit:
a. lim 5
b. lim x
x 3
x3
c. lim x 4
x 3
Note: In all three cases, lim f ( x) _
x c
So, we can evaluate the limits by substituting c for x.
Theorem: Basic Limits
Let b and c be real numbers, and l
MAT 1730
5.4 The Fundamental Theorem of Calculus
Theorem The Fundamental Theorem of Calculus: If f is a continuous function on [a, b] and F is an
antiderivative of f on [a, b] , then
b
a
f ( x)dx F ( x)
b
a
F (b) F (a) .
Proof:
Let : If f be a continuous
MAT 1730
3.7 Related Rates.
The Chain Rule always applies when you have one variable, and you are taking the derivative with respect to a
different variable than the one used in the problem.
e.g. Find the stated derivative:
d
a.
ln y
dx
b.
d
w2 3w 2
dx
MAT 1730
3.1: The Derivative and Tangent Line Problem
How do you find the slope of the line that goes through the points ( x1 , y1 ) and ( x2 , y2 ) ?
def: The tangent line to a circle at a given point is the line through that point that is perpendicular
MAT 1730
3.2: Rates of Change & Basic Differentiation Rules
Find the derivative of the functions:
a. f ( x) c , for some real number c.
b. f ( x) x
Theorem The Constant Rule:
d
c 0
dx
Special Case of the Power Rule:
d
x 1
dx
The Power Rule: If f ( x) x n
#4
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DORLANDO
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Starbucks for Life: Summer Edition
Official Rules
NO PU RC H AS E N EC E S S AR Y . A P U R CH AS E O R P AY M E NT O F AN Y KI ND W ILL
NO T IN CR E AS E Y O U R CH AN C E S O F W I NN ING .
1. Eligibility: Starbucks for Life: Summer Edition (the Promoti
Section 7.6
Moments, Centers of Mass, and Centroids
Moments and Center of Mass: One Dimensional System (487)
Let the point masses
be located at
on the x axis.
1. The moment about the origin is
2. The center of mass is
where
is the total mass of the system