Copyright Sautter 2003

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Thanks for your patience!Walt S.

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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

IF WE ADD 10 TO THE LEFT SIDE WE MUST

ADD 10 TO THE RIGHT

IF WE MULTIPLY THE LEFTSIDE BY 5 WE MUST

MULTIPLY THE RIGHT BY 5

RULE 1 – A VALUE DIVIDED BY ITSELF EQUALS 1

RULE 2 – OPERATE ON BOTH SIDES EQUALLY

ALGEBRA & EQUATIONS

RULE 3 – OPERATION IN PARENTHESES ARE DONE FIRST

THE PARENTHESES TERMS (5 + 5) ARE

ADDED FIRST

THE PARENTHESESTERMS (22 – 7) ARE

SUBTRACTED FIRST

ALGEBRA & EQUATIONSRULE 4 – VALUES CAN BE DISTRIBUTED

THROUGH TERMS IN PARENTHESES

EACH TERM IN THEPARENTHESES MUSTBE MULTIPLIED BY 4

ALL TERMS MUST BEMULTIPLIED BY

EACHOTHER THEN ADDED

ALGEBRA & EQUATIONS

RULE 5 – WHEN A NUMERATOR TERM ISDIVIDED BY A DENOMINATOR TERM, THE

DENOMINATOR IS INVERTED AND MULTIPLIEDBY THE NUMERATOR TERM.

invertmultiple

Distribute terms

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.

• THESE RULES CAN BE IMPLEMENTED PRACTICALLY USING SIMPLIFIED PROCEDURES (KEEP IN MIND THE REASON THAT THESE PROCEDURES WORK IS BECAUSE OF THE ALGEBRAIC RULES).

SOLVING ALGEBRAIC EQUATIONS

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 fmust be 10

The value of f with the solvedequation is 10 !

The solution to the quadratic gives the valuesof X when the value of Y is zero.

(the roots of the equation)

QuadraticEquations Have TwoAnswers

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.

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

X THE VERTICALVARIABLE

THE HORIZONTALVARIABLE

SLOPE

VERTICALINTERCEPT

POINTb

rise

run

SLOPE = RISE / RUNSLOPE = Y / X

Y

XA constant

A positive power

other than 1 or zero

The slope is always changing (variable)

SLOPE ?

CONSTANT SLOPE ?POSITIVE OR NEGATIVE?

CONSTANT SLOPE ?POSITVE OR NEGATIVE ?

SLOPE = 0

CONSTANT SLOPE ?POSITIVE OR NEGATIVE ?

SLOPES & RATES

TIME TIME

TIME TIME

SLOPE = RISE / RUN

SLOPE IS NEGATIVESLOPE IS CONSTANT

SLOPE IS NEGATIVESLOPE IS VARIABLE

SLOPE IS POSITIVESLOPE IS VARIABLE

SLOPE OF A TANGENT LINE TO A POINT = INSTANTANEOUS RATE

GRAPH 1 GRAPH 2

GRAPH 3 GRAPH 4

DISPLACEMENT

DISPLACEMENT

DISPLACEMENT

DISPLACEMENT

DISPLACEMENT Time

VELOCITY

Time

ACCELERATION Time

S

t

t

v

Slope of a tangent drawn to a point ona displacement vs time graph gives

the instantaneous velocity at that point

Slope of a tangent drawn to a point ona velocity vs time graph gives the

instantaneous acceleration at that point

Y

X

X1 X2

AREA UNDER THE CURVEFROM X1 TO X2

Area = Y X (SUM OF THE BOXES)

WIDTH OF EACH BOX = X

AREA MISSED - INCREASINGTHE NUMBER BOXES WILL

REDUCE THIS ERROR!

AS THE NUMBER OF BOXESINCREASES, THE ERROR

DECREASES!

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 SIMPLEEXPONENTIAL EQUATIONS

THE DERIVATIVE OF A EQUATION GIVES ANOTHEREQUATION 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 thePower times

The equation

Subtract oneFrom the

power

dy/dx is the mathematicalSymbol for the derivative

APPLYING THE DERIVATIVE FORMULA

GIVEN THEEQUATION

FORMAT TO FIND THE

DERIVATIVE

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.

DerivativesCan be used

To find:Velocity,

Acceleration,AngularVelocity,Angular

Acceleration,Etc.

APPLYING THE DERIVATIVE FORMULA

The derivatives of equations having more than one term canbe 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 ANTIDERIVATIVEINTEGRATION IS THE REVERSE PROCESS OF

FINDING THE DERIVATIVE. IT CAN ALSO BE USEDTO FIND THE AREA UNDER A CURVE.

THE GENERAL FORMAT FOR FINDING THE INTEGRAL OF A SIMPLE POWER RELATIONSHIP

ADD ONE TO THE POWER

DIVIDE THEEQUATION

BY THE N + 1

ADD A CONSTANT

is the symbolfor integration

APPLYING THE INTEGRAL FORMULA

GIVEN THEEQUATION

FORMAT TO FIND THE INTEGRAL

Integration can be used to find area under a curve betweentwo points. Also, if the original equation is a derivate, then

the equation from which the derivate came can be determined.

APPLYING THE INTEGRAL FORMULAFind 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 arecalled the limits.

each of the limits isplaced in the integratedequation and the resultsof each calculation aresubtracted (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 = +

y

x

+

+

-

-

Quadrant IQuadrant II

Quadrant III Quadrant IV

0 o

90 o

180 o

270 o

360 o

Rectangular Coordinates

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

y

x

+

+

-

-

Quadrant IQuadrant II

Quadrant III Quadrant IV

0 radians radians

3/2 radians

2 radians

/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 WAYTO 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

A RIGHT TRIANGLE

A & B

B

CCCA

900

900

900+ + = 1800

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)

A

B

C

Sin = A / C

Cos = B / C

Tan = A / B

AC

B

A

BA RIGHT TRIANGLE

C

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.

y

x

+

+

-

- 0 radians radians

3/2 radians

2 radians

Quadrant III

Quadrant IV

Quadrant I

Quadrant II

Sin Cos Tan + + +

+ - -- - +

- + -

/2 radians90 o

0 o

180 o

270 o

360 o

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 1As the decimal is moved to the left

The power of 10 increases onevalue for each decimal place moved

Any number to theZero power = 1

RULE 2As the decimal is moved to the right

The power of 10 decreases onevalue for each decimal place moved

Any number to theZero power = 1

RULE 3When scientific numbers are multiplied

The powers of 10 are added

RULE 4When scientific numbers are divided

The powers of 10 are subtracted

RULE 5When scientific numbers are raised to powers

The powers of 10 are multiplied

RULE 6Roots of scientific numbers are treated as fractional

powers. The powers of 10 are multiplied

RULE 7When scientific numbers are added or subtracted The powers of 10 must be the same for each term.

Powers of 10 areDifferent. ValuesCannot be added !

Power are now theSame and values

Can be added.

Move the decimalAnd change the power

Of 10

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.