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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.