Formula Sheet for Intermediate Algebra Final Exam
Properties of Exponents
n
m
1. a a a
Quadratic Formula
n m
x
an
2. m a n m
a
3. (a n ) m a nm
n
m
p
6. b p
np
mp
p
a np
mp
b
1
p
b
b
2.
n
3.
m n
n
n
4.
n
m
n b m b
Absolute Value Inequalities
m
n
b mn b
U-Substitution
Since the above integration formulas are limiting our ability to
perform such operations (remember there is no product or quotient
rule) we must utilize a method called u-substitution to produce
expressions similar to the ones listed above.
Area under the Curve
&
the Fundamental Theorem of Calculus
Definition of a definite Integral (integral with integration limits)
b
a
f ( x)dx
is called a definite integral with integration limits of
x = a and x = b
Evaluating a definite integral:
3 1
3x
3
Exponential Growth or Decay
One of the most powerful applications of the process of
Integrations (Anti-differentiation) is the one of converting a Rate
of Change equation or expression into a total amount function.
Many processes in nature, science, or fi
LHopitals Rule
LHopitals Rule is a technique used to evaluate limits of the
following forms.
0
, , and several other variations of this
0
type.
LHopitals Rule states that
Lim
x c
f ( x)
f ' ( x)
= Lim
if Lim f ( x) = Lim g ( x) = 0 or
x c
g ( x) xc g ('
The Mean Value Theorem
The Mean Value Theorem represents yet another application of the
Derivative.
Let f(x) be continues on [a,b] and differentiable on (a,b) then there
exists some number c in [a,b] such that:
f (b ) f ( a )
f ' (c ) =
ba
In other words
Exploring Function Behavior!
Most of us are familiar with the procedure of examining the graph of a
function. In algebra we were taught to view a graph from left to right
discussing its behavior i.e. noting the intervals where it appears to be
increasing,
Optimization
The term OPTIMIZATION refers to the process of maximizing or
minimizing application problems. For example, we may want to
determine the maximum area we can fence in with a certain
amount of fencing material or the dimensions, which would
maxi
Derivatives of Trigonometric Functions
Derivatives of sin(x) and cos(x)
The derivatives of the six basic trigonometric functions
are relatively straight forward.
First of all the derivative of the
While the derivative of
sin(x) = cos(x).
cos(x) = sin(x).
Higher-Order Derivatives
The process of differentiation can be applied to an expression more
then one time and successive differentiations require a special
notation identifying the Order or level of differentiation. One of
these notations is the (prime)