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Algebra and Precalculus Review
Aug 21, 2011
Things you Should know for Differential Calculus
8/12/2011
Formulas and Rules you are expected to know.
Zero Product rule:
or
0
=
AB
⇔
0
=
A
0
=
B
.
Note: The symbol
⇔
means
if and only if.
Trichotomy:
Given an arbitrary real number, say
x
, then
x
is either positive (
), negative
(
), or zero (
).
0
>
x
0
<
x
0
=
x
The definition of absolute value _____________________
The Quadratic formula:
_________________________
The Pythagorean Theorem:
________________________
Items from geometry
geometric description of line in a plane.
geometric description of a circle
Formulas for the area and perimeter of the following planar regions
Rectangle, Parallelogram, Trapezoid, Triangle, Circle
Formulas for the surface area and volume of
spheres, rectangular prisms,
circular cylinders
Trigonometry
The definition of Sine, Cosine, and the other 4 trigonometric functions.
Right triangle trigonometry
Selected identities
equivalent notation
1
cos
sin
2
2
=
+
A
A
( ) ( )
1
cos
sin
2
2
=
+
A
A
A
A
2
2
sec
1
tan
=
+
( ) ( )
2
2
sec
1
tan
A
A
=
+
A
A
2
2
csc
cot
1
=
+
( ) ( )
2
2
csc
cot
1
A
A
=
+
Double Angle formulas
)
)(cos
(sin
2
2
sin
A
A
A
=
A
A
A
2
2
sin
cos
2
cos
−
=
1
)
(cos
2
2
−
=
A
A
2
sin
2
1
−
=
and their offspring, the power reduction formulas
2
2
cos
1
sin
2
A
A
−
=
2
2
cos
1
cos
2
A
A
+
=
Law of Cosines:
(draw the triangle and label the appropriate
angles and legs per the formula)
B
ac
c
a
b
cos
2
2
2
2
−
+
=
Law of Sines:
c
C
b
B
a
A
sin
sin
sin
=
=
(draw the triangle and label the appropriate
angles and legs per the formula)
A “simplified” expression is one:

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(1)
in which all like terms are collected,
(2)
which does not contain negative exponents,
(3)
in which common factors are reduced,
(4)
where the result is not a complex fraction. i.e. Quotients do not contain fractions in the
numerator or in the denominator.
(5)
in which radical expressions are in reduced form.
Make sure that you can derive the simplified version in each of these 5 examples.
Example 1.
violates (1)
Simplified expression:
)
4
)(
2
3
(
)
4
5
)(
4
3
(
2
x
x
x
x
−
+
+
−
15
4
24
2
+
+
−
x
x
Example 2.
2
2
2
)
5
(
)
5
2
)(
1
3
(
)
3
)(
5
(
x
x
x
x
x
x
+
+
−
−
+
the numerator violates (1)
Simplified expression:
2
2
2
)
5
(
5
2
3
x
x
x
x
+
+
+
−
Example 3.
2
3
2
1
2
16
2
1
−
−
−
x
x
violates (2) and (3)
Simplified expression:
2
3
8
2
1
x
x
−
Example 4.
2
3
1
3
2
)
4
3
(
)
4
(
)
4
3
(
3
1
x
x
x
x
−
−
−
−
violates (1) and (4)
Simplified expression:
2
3
2
)
4
3
(
3
8
3
x
x
x
−
+
Example 5.
4
3
72
y
x
Simplified expression
2
2
6
y
x
x
Each simplified expression above is consistent with the simplification requirements, however, further
algebraic manipulation may be required depending on what one wishes to do with the expression.
For example if one were interested in finding the values of

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