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Copyright Sautter 2003
ALGEBRA & EQUATIONS
• THE USE OF BASIC ALGEBRA REQUIRES ONLY A FEW
FUNDAMENTAL RULES WHICH ARE USED OVER AND
OVER TO REARRANGE AND SOLVE EQUATIONS.
• (1) ANY NUMBER DIVIDED BY ITSELF IS EQUAL TO
ONE.
• (2) WHAT IS EVER DONE TO ONE SIDE OF AN
EQUATION MUST BE DONE EQUALLY TO THE OTHER
SIDE.
• (3) ADDITIONS OR SUBTRACTIONS WHICH ARE
ENCLOSED IN PARENTHESES ARE GENERALLY
CARRIED OUT FIRST.
• (4) WHEN VALUES IN PARENTHESES ARE MULTIPLIED
OR DIVIDED BY A COMMON TERM EACH CAN BE
MULTIPLIED OR DIVIDED SEPARATELY BEFORE
ADDING OR SUBTRACTING THE GROUPED TERMS.
ALGEBRA & EQUATIONS
RULE 1 – A VALUE DIVIDED BY ITSELF EQUALS 1
RULE 2 – OPERATE ON BOTH SIDES EQUALLY
IF WE ADD 10 TO THE
LEFT SIDE WE MUST
ADD 10 TO THE RIGHT
IF WE MULTIPLY THE LEFT
SIDE BY 5 WE MUST
MULTIPLY THE RIGHT BY 5
ALGEBRA & EQUATIONS
RULE 3 – OPERATION IN PARENTHESES ARE DONE FIRST
THE PARENTHESES
TERMS (5 + 5) ARE
ADDED FIRST
THE PARENTHESES
TERMS (22 – 7) ARE
SUBTRACTED FIRST
ALGEBRA & EQUATIONS
RULE 4 – VALUES CAN BE DISTRIBUTED
THROUGH TERMS IN PARENTHESES
EACH TERM IN THE
PARENTHESES MUST
BE MULTIPLIED BY 4
ALL TERMS MUST BE
MULTIPLIED BY
EACHOTHER THEN ADDED
ALGEBRA & EQUATIONS
RULE 5 – WHEN A NUMERATOR TERM IS
DIVIDED BY A DENOMINATOR TERM, THE
DENOMINATOR IS INVERTED AND MULTIPLIED
BY THE NUMERATOR TERM.
invert
Distribute terms
multiple
SOLVING ALGEBRAIC EQUATIONS
• SOLVING AN ALGEBRAIC EQUATION REQUIRES
THAT THE UNKNOWN VARIABLE BE ISOLATED
ON THE LEFT SIDE OF THE EQUAL SIGN IN THE
NUMERATOR POSITION AND ALL OTHER TERMS
BE PLACED ON THE RIGHT SIDE OF THE EQUAL
SIGN.
• THIS MOVEMENT OF TERMS FROM LEFT TO
RIGHT AND FROM NUMERATOR TO
DENOMINATION AND BACK, IS ACCOMPLISHED
USING THE BASIC RULES OF ALGEBRA WHICH
WERE PREVIOUSLY DISCUSSED.
SOLVING ALGEBRAIC EQUATIONS
•
THESE RULES CAN BE IMPLEMENTED PRACTICALLY
USING SIMPLIFIED PROCEDURES (KEEP IN MIND THE
REASON THAT THESE PROCEDURES WORK IS BECAUSE
OF THE ALGEBRAIC RULES).
PROCEDURE 1 – WHEN A TERM WITH A PLUS OR MINUS
SIGN IS MOVED FROM ONE SIDE OF THE EQUATION
TO THE OTHER, THE SIGN IS CHANGED.
SOLVING ALGEBRAIC EQUATIONS
PROCEDURE 2 – WHEN A TERM IS MOVED FROM
THE DENOMINATOR ACROSS AN EQUAL SIGN TO THE
OTHER SIDE OF THE EQUATION IT IS PLACED IN THE
NUMERATOR. LIKEWISE, WHEN A TERM IS MOVED
FROM NUMERATOR ON ONE SIDE IT IS PLACED IN THE
DENOMINATOR ON THE OTHER SIDE.
Distribute g and
Multiple each side
By -1
Solutions to algebraic equations can be checked by
inserting simple number values. Avoid using 1 since it
is a special case value.
Let a =4, b=6,
c = 2, e = 3
and g =5
The value of f
must be 10
The value of f
with the solved
equation is 10 !
Calculations often require the use of the quadratic equation.
It is used to solve equations containing a squared, a first power
and a zero power (constant) term all in the same equation.
The solution to the quadratic gives the values
of X when the value of Y is zero.
(the roots of the equation)
Quadratic
Equations
Have Two
Answers
USING THE QUADRATIC EQUATION
• Here is an example using the quadratic equation. In this
equation 4x2 is the squared term, 0.0048X is the first
power term and zero power term is –3.2 x 10-4 (a
constant)
• 4X2 +0.0048X – 3.2 x 10-4 = 0 this equation cannot be
solved easily by inspection and requires the quadratic
formula:
• Using the form aX2 + bX + c = 0 the formula is:
• ( - b + b2 – 4ac )/ 2a
• In the given equation: a = 4, b = 0.0048 and c = – 3.2 x 10-4
• (-0.0048 + (0.0048)2 – 4(4)(– 3.2 x 10-4 )) / 2(4) = 0.0083
-0.0095
• Note: every quadratic has two answers.
GRAPHS AND EQUATIONS
• GRAPHS CAN BE CONSIDERED AS A PICTURE OF
AN EQUATION SHOWING AN ARRAY OF X AND Y
VALUES WHICH WERE CALCULATED FROM THE
EQUATION.\
• WE WILL LOOK AT TWO DIFFERENT KINDS OF
GRAPHS, LINEAR (STRAIGHT LINE) AND CURVED.
• LINEAR GRAPHS ARE DESCRIBED BY THE
GENERAL EQUATION: Y = mX + b
• CURVED GRAPHS ARE DESCRIBED BY THE
GENERAL EQUATION: Y = Kx n
• ALTHOUGH GRAPHS CAN BE REPRESENTED BY
MANY OTHER EQUATIONS, WE WILL LOOK AT
ONLY THESE TWO BASIC RELATIONSHIPS IN
DETAIL
Y
VERTICAL
THE HORIZONTAL INTERCEPT
VARIABLE
POINT
b
rise
run
X
THE VERTICAL
VARIABLE
SLOPE = RISE / RUN
SLOPE =  Y / X
SLOPE
A positive
power
other than
1 or zero
Y
A constant
X
The slope is always changing (variable)
SLOPES & RATES
SLOPE = RISE / RUN
D
I
S
P
L
A
C
E
M
E
N
T
GRAPH 1
SLOPE =? 0
SLOPE
D
I
S
P
L
A
C
E
M
E
N
T
TIME
D
I
S
P
L
A
C
E
M
E
N
T
GRAPH 2
SLOPE
CONSTANT
IS NEGATIVE
SLOPE ?
POSITIVE
SLOPE IS
ORCONSTANT
NEGATIVE ?
TIME
SLOPE
CONSTANT
IS NEGATIVE
SLOPE ?
POSITVE
SLOPE OR
IS VARIABLE
NEGATIVE ?
GRAPH 3
D
I
S
P
L
A
C
E
M
E
N
T
SLOPE IS POSITIVE
CONSTANT
SLOPE ?
SLOPE IS
POSITIVE
ORVARIABLE
NEGATIVE?
GRAPH 4
TIME
TIME
SLOPE OF A TANGENT LINE TO A POINT = INSTANTANEOUS RATE
Slope of a tangent drawn to a point on
a displacement vs time graph gives
the instantaneous velocity at that point
D
I
S
P
L
A
C
E
M
E
N
T
S
Time
t
V
E
L
O
C
I
T
Y
v
Time
t
A
C
C
E
L
E
R
A
T
I
O
N
Time
Slope of a tangent drawn to a point on
a velocity vs time graph gives the
instantaneous acceleration at that point
AREA UNDER THE CURVE
FROM X1 TO X2
Area =  Y  X (SUM OF THE BOXES)
AREA
MISSED
- INCREASING
AS THE
NUMBER
OF BOXES
THE
NUMBER THE
BOXES
WILL
INCREASES,
ERROR
REDUCE
THIS ERROR!
DECREASES!
Y
WIDTH OF EACH BOX =  X
X1
X2
X
MATHEMATICAL SLOPES & AREAS
• IF THE EQUATION FOR A GRAPH IS KNOWN THE
SLOPE OF THAT GRAPH LINE CAN BE FOUND
MATHEMATICALLY USING A PROCESS CALLED A
DERIVATIVE.
• IF THE EQUATION FOR A GRAPH IS KNOWN THE
AREA UNDER THE CURVE CAN BE FOUND USING A
PROCESS CALLED INTEGRATION.
• IF THE EQUATION DESCRIBING THE SLOPE OF A
GRAPH IS KNOWN THE EQUATION FOR THE GRAPH
CAN BE FOUND USING INTEGRATION.
• THE NEXT FRAMES WILL SHOW ELEMENTARY
DERIVATIVES AND INTEGRALS WITHOUT
PROVIDING ANY FORMAL MATHEMATICAL PROOF.
IF PROOF IS DESIRED SEE A CALCULUS TEXT!
FINDING DERIVATIVES OF SIMPLE
EXPONENTIAL EQUATIONS
THE DERIVATIVE OF A EQUATION GIVES ANOTHER
EQUATION WHICH ALLOWS THE SLOPE OF THE
ORIGINAL EQUATION TO FOUND AT ANY POINT.
THE GENERAL FORMAT FOR FINDING THE
DERIVATIVE OF A SIMPLE POWER RELATIONSHIP
Multiple the
Power times
The equation
dy/dx is the mathematical
Symbol for the derivative
Subtract one
From the
power
APPLYING THE DERIVATIVE FORMULA
GIVEN THE
EQUATION
FORMAT TO
FIND THE
DERIVATIVE
Derivatives
Can be used
To find:
Velocity,
Acceleration,
Angular
Velocity,
Angular
Acceleration,
Etc.
Using the derivative equation we can find the slope of the y = 5 x3
equation at any x point. For example, the slope at x = 2 is
Slope = 15 x 22 = 60. At x = 5, slope = 15 x 52 = 375.
APPLYING THE DERIVATIVE FORMULA
The derivatives of equations having more than one term can
be found by finding the derivative of each term in succession.
Recall that the term 3t is actually 3t1 and the term 6 is 6t0.
Also, any term to the zero power equals one
INTEGRATION – THE ANTIDERIVATIVE
INTEGRATION IS THE REVERSE PROCESS OF
FINDING THE DERIVATIVE. IT CAN ALSO BE USED
TO FIND THE AREA UNDER A CURVE.
THE GENERAL FORMAT FOR FINDING THE
INTEGRAL OF A SIMPLE POWER RELATIONSHIP
ADD ONE
TO THE
POWER
 is the symbol
for integration
 
DIVIDE THE
EQUATION
BY THE N + 1
ADD A
CONSTANT
APPLYING THE INTEGRAL FORMULA
GIVEN THE
EQUATION
FORMAT TO
FIND THE
INTEGRAL
 

Integration can be used to find area under a curve between
two points. Also, if the original equation is a derivate, then
the equation from which the derivate came can be determined.
APPLYING THE INTEGRAL FORMULA
Find the area between x = 2 and x = 5 for the equation y = 5X3.
First find the integral of the equation as shown on the previous
frame. The integral was found to be 5/4 X4 + C.
The values 5 and 2 are
called the limits.
each of the limits is
placed in the integrated
equation and the results
of each calculation are
subtracted (lower limit
from upper limit)
MEASURING DIRECTION & POSITION
• RECTANGULAR COORDINATES USE X,Y POINTS
TO INDICATE DISPLACEMENTS AND
DIRECTIONS.
• POLAR COORDINATES USE MAGNITUDES
(LENGTHS) AND ANGULAR DIRECTION. THE
ANGULAR DIRECTION MAY BE EXPRESSED IN
DEGREES OR RADIANS.
• DIRECTIONS CAN ALSO BE INDICATED IN
GEOGRAPHIC TERMS SUCH AS NORTH, SOUTH,
EAST AND WEST.
• OFTEN, GEOGRAPHIC MEASURES AND
ANGULAR MEASURES ARE COMBINED TO
INDICATE DIRECTION.
Up = +
Down = -
Right = +
Left = +
Rectangular Coordinates
Quadrant II
180 o
90 o
y
+
-
Quadrant I
o
x 0
360 o
Quadrant IV
+
Quadrant III
270 o
RADIANS = ARC LENGTH / RADIUS LENGTH
CIRCUMFERENCE OF A CIRCLE = 2  x RADIUS
RADIANS IN A CIRCLE = 2  R / R
1 CIRCLE = 2  RADIANS = 360O
1 RADIAN = 360O / 2  = 57.3O
/2 radians
y
Quadrant II +
Quadrant I
 radians
-
+
x 0 radians
2 radians
Quadrant IV
Quadrant III
3/2  radians
NOTICE THAT THESE DIRECTIONS ARE NOT PRECISE !
GEOGRAPHIC DIRECTIONS
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
GEOGRAPHIC DIRECTIONS ARE OFTEN EQUATED TO ANGULAR
MEASURES AS FOLLOWS:
EAST (E) = 0 DEGREES
EAST NORTHEAST (ENE) = 22.5 DEGREES
NORTHEAST (NE) = 45 DEGREES
NORTH NORTHEAST (NNE) = 67.5 DEGREES
NORTH (N) = 90 DEGREES
NORTH NORTHWEST (NNW) = 112.5 DEGREES
NORTHWEST (NW) = 135 DEGREES
WEST NORTHWEST (WNW) = 157.5 DEGREES
WEST (W) = 180 DEGREES
WEST SOUTH WEST (WSW) = 202.5
SOUTH WEST (SW) =225 DEGREES
SOUTH SOUTH WEST (SSW) = 247.5 DEGREES
SOUTH (S) = 270 DEGREES
SOUTH SOUTHEAST (SSE) = 292.5 DEGREES
SOUTHEAST (SE) = 315 DEGREES
EAST SOUTH EAST (ESE) = 337.5 DEGREES
500 NORTH OF EAST
250 WEST OF SOUTH
-450
(ANOTHER WAY
TO MEASURE
ANGLES)
TRIGNOMETRY
• TRIGNOMETRIC RELATIONSHIPS ARE BASES ON
THE RIGHT TRIANGLE (A TRIANGLE CONTAINING A
900 ANGLE). THE MOST FUNDAMENTAL CONCEPT IS
THE PYTHAGOREAN THEOREM (A2 + B2 = C2)
WHERE A AND B ARE THE SHORTER SIDES (THE
LEGS) OF THE TRIANGLE AND C IS THE LONGEST
SIDE CALLED THE HYPOTENUSE.
• RATIOS OF THE SIDES OF THE RIGHT TRIANGLE
ARE GIVEN NAMES SUCH AS SINE, COSINE AND
TANGENT. DEPENDING ON THE ANGLE BETWEEN A
LEG (ONE OF THE SHORTER SIDES) AND THE
HYPOTENUSE (THE LONGEST SIDE), THE RATIO OF
SIDES FOR A PARTICULAR ANGLE ALWAYS HAS THE
SAME VALUE NO MATTER WHAT SIZE THE
TRIANGLE.
C

CC
A
900

BB
A RIGHT TRIANGLE
 +  + 900 = 1800
A & B
TRIG FUNCTIONS
• THE RATIO OF THE SIDE OPPOSITE THE ANGLE AND
THE HYPOTENUSE IS CALLED THE SINE OF THE ANGLE.
THE SINE OF 30 0 FOR EXAMPLE IS ALWAYS ½ NO
MATTER HOW LARGE OR SMALL THE TRIANGLE. THIS
MEANS THAT THE OPPOSITE SIDE IS ALWAYS HALF AS
LONG AS THE HYPOTENUSE IF THE ANGLE IS 30 0. (30 0
COORESPONSES TO 1/12 OF A CIRCLE OR ONE SLICE OF
A 12 SLICE PIZZA!)
• THE RATIO OF THE SIDE ADJACENT TO THE ANGLE AND
THE HYPOTENUSE IS CALLED THE COSINE. THE COSINE
OF 60 0 IS ALWAYS ½ WHICH MEANS THIS TIME THE
ADJACENT SIDE IS HALF AS LONG AS THE HYPOTENUSE.
(60 0 REPRESENTS 1/6 OF A COMPLETE CIRCLE, ONE
SLICE OF A 6 SLICE PIZZA)
• THE RATIO OF THE SIDE ADJACENT TO THE ANGLE AND
THE SIDE OPPOSITE THE ANGLE IS CALLED THE
TANGENT. IF THE ADJACENT AND THE OPPOSITE SIDES
ARE EQUAL, THE RATIO (TANGENT VALUE) IS 1.0 AND
THE ANGLE IS 45 0 ( 45 0 IS 1/8 OF A FULL CIRCLE)
Sin  = A / C
CC
A
A
A
Cos  = B / C
Tan  = A / B

BB
A RIGHT TRIANGLE
Trig functions
• Right triangles may be drawn in any one of four quadrants.
• Quadrant I encompasses from 0 to 90 degrees (1/4 of a circle). It
lies between the +x axis and the + y axis (between due east and
due north).
• Quadrant II is the area between 90 and 180 degrees ( the next ¼
circle in the counterclockwise direction). It lies between the +y
and the –x axis (between due north and due west).
• Quadrant III is the area between 180 and 270 degrees (the next ¼
circle in the counterclockwise direction). It lies between the –x and
the –y axis (between due west and due south).
• Quadrant IV encompasses from 270 to 360 degrees ( the final ¼
circle). It lies between the –y and the +x axis (between due south
and due east).
• The signs of the trig functions change depending upon in which
quadrant the triangle is drawn.
/2 radians
y
90 o
+



-

180 o
 radians
Quadrant I
Sin
+
Cos 
+
Quadrant II
+
-
Quadrant III
-
+
-
Quadrant IV
0o
+
x 0 radians
2 radians
360 o
270 o
3/2  radians
Tan 
+
+
In science, we often encounter very large and very
small numbers. Using scientific numbers makes
working with these numbers easier
Scientific numbers use powers of 10
RULE 1
As the decimal is moved to the left
The power of 10 increases one
value for each decimal place moved
Any number to the
Zero power = 1
RULE 2
As the decimal is moved to the right
The power of 10 decreases one
value for each decimal place moved
Any number to the
Zero power = 1
RULE 3
When scientific numbers are multiplied
The powers of 10 are added
RULE 4
When scientific numbers are divided
The powers of 10 are subtracted
RULE 5
When scientific numbers are raised to powers
The powers of 10 are multiplied
RULE 6
Roots of scientific numbers are treated as fractional
powers. The powers of 10 are multiplied
RULE 7
When scientific numbers are added or subtracted
The powers of 10 must be the same for each term.
Powers of 10 are
Different. Values
Cannot be added !
Move the decimal
And change the power
Of 10
Power are now the
Same and values
Can be added.
LOGARITHMS
• A LOGARITHM (LOG) IS A POWER OF 10. IF A NUMBER IS
WRITTEN AS 10X THEN ITS LOG IS X.
• FOR EXAMPLE 100 COULD BE WRITTEN AS 102
THEREFORE THE LOG OF 100 IS 2.
• IN CHEMISTRY CALCULATIONS OFTEN SMALL NUMBERS
ARE USED LIKE .0001 OR 10-4. THE LOG OF .0001 IS
THEREFORE –4.
• FOR NUMBERS THAT ARE NOT NICE EVEN POWERS OF 10
A CALCULATOR IS USED TO FIND THE LOG VALUE. FOR
EXAMPLE THE LOG OF .00345 IS –2.46 AS DETERMINED
BY THE CALCULATOR.
• LOGARITHMS DO NOT ALWAYS USE POWERS OF 10.
ANOTHER COMMON NUMBER USED INSTEAD OF 10 IS
2.71 WHICH IS CALLED BASE e. WHEN THE LOGARITHM IS
THE POWER OF e IT IS CALLED A NATURAL LOG AND THE
SYMBOL USED IN Ln RATHER THAN LOG.
LOGARITHMS
• SINCE LOGS ARE POWERS OF 10 THEY ARE
USED JUST LIKE THE POWERS OF 10
ASSOCIATED WITH SCIENTIFIC NUMBERS.
• WHEN LOG VALUES ARE ADDED, THE NUMBERS
THEY REPRESENT ARE MULTIPLIED.
• WHEN LOG VALUES ARE SUBTRACTED, THE
NUMBERS THEY REPRESENT ARE DIVIDED
• WHEN LOGS ARE MULTIPLIED, THE NUMBERS
THEY REPRESENT ARE RAISED TO POWERS
• WHEN LOGS ARE DIVIDED, THE ROOTS OF
NUMBERS THEY REPRESENT ARE TAKEN.