Basic algebra, trig and calculus needed for physics. **More good stuff available at: www.wsautter.com and http://www.youtube.com/results?search_query=wnsautter&aq=f

1. BASIC MATH FOR PHYSICS Algebra, Trig, Geometry & Derivatives Copyright Sautter 2003

2. The next slide is a quick promo for my books after which the presentation will begin Thanks for your patience! Walt S. [email_address] More stuff at: www. wsautter .com

3. Books available at: www. wsautter .com www.smashwords.com www.amazon.com www.bibliotastic.com www.goodreads.com Walt’s Books for Free!

5. ALGEBRA & EQUATIONS 10/10 =1 X /X = 1 Y /Y =1 X + 5 = Y 10 + + 10 X + 15 = Y + 10 IF WE ADD 10 TO THE LEFT SIDE WE MUST ADD 10 TO THE RIGHT X + 5 = Y IF WE MULTIPLY THE LEFT SIDE BY 5 WE MUST MULTIPLY THE RIGHT BY 5 X + 5 = Y 5 x ( ) 5 x 5X + 25 = 5Y RULE 1 – A VALUE DIVIDED BY ITSELF EQUALS 1 RULE 2 – OPERATE ON BOTH SIDES EQUALLY

6. ALGEBRA & EQUATIONS RULE 3 – OPERATION IN PARENTHESES ARE DONE FIRST Y = ( 5 + 4 ) ( X + 2) THE PARENTHESES TERMS (5 + 5) ARE ADDED FIRST Y = 9 ( X + 2 ) Y = 9 X + 18 S = T ( 22 - 7 ) THE PARENTHESES TERMS (22 – 7) ARE SUBTRACTED FIRST 2 S = 225 T S = 15 T 2

7. ALGEBRA & EQUATIONS RULE 4 – VALUES CAN BE DISTRIBUTED THROUGH TERMS IN PARENTHESES Y = 4 ( T + 15 ) EACH TERM IN THE PARENTHESES MUST BE MULTIPLIED BY 4 Y = 4 T + 60 Y = ( R + 2 ) ( R - 3 ) ALL TERMS MUST BE MULTIPLIED BY EACHOTHER THEN ADDED Y = ( R x R ) - 3 R + 2 R - 6 Y = R - 1R - 6 2 Y = R - R - 6 2

8. = ALGEBRA & EQUATIONS RULE 5 – WHEN A NUMERATOR TERM IS DIVIDED BY A DENOMINATOR TERM, THE DENOMINATOR IS INVERTED AND MULTIPLIED BY THE NUMERATOR TERM. = a a invert multiple Distribute terms a ------------------ b / ( c + d ) ( c + d ) ------------------ b ( c + d ) ------------------ b c + d ------------------ b x a x a

11. 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. B M K A C B D ----- = ------ A C D = ----------- x B ----- = --- F x K G M N = ---------- F G x M N x K

12. = a + b / c = Some Common Algebraic Equalities a + b -------------- = c a b + c c ----- ----- ---- 1 c ( a + b ) a b + c ----- a + b ------------ c = / a + b -------------- = c ( a + b ) / c

13. Solve for f e = Distribute g and Multiple each side By -1 f + Solving an Algebraic Equation g ( ) a + b -------------- = c e f + g ----- - f g ----- a + b -------------- = c - f g ----- a + b -- -------------- c e f + g ----- a + b -- -------------- c g = e a + b + -------------- c -- ( ) g

14. Checking Algebraic Solutions 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 ! a + b -------------- = c e f + g ----- 4 + 6 -------------- = 2 3 f + 5 ----- g = e a + b + -------------- c -- ( ) g f f 5 = 3 4 + 6 + -------------- 2 -- ( ) 5 x = 10

15. The Quadratic Equation x = 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 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. X + X + = y 2 a b c - b b - 2 a + - 2 ------------------------- / ------------------------- a c

18. STRAIGHT LINE GRAPHS Y = m X + b THE VERTICAL VARIABLE THE HORIZONTAL VARIABLE SLOPE VERTICAL INTERCEPT POINT b rise run SLOPE = RISE / RUN SLOPE =  Y /  X Y X

19. Curved graphs Y X A constant A positive power other than 1 or zero The slope is always changing ( variable) Y = k X n

20. 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 NEGATIVE SLOPE IS CONSTANT SLOPE IS NEGATIVE SLOPE IS VARIABLE SLOPE IS POSITIVE SLOPE IS VARIABLE SLOPE OF A TANGENT LINE TO A POINT = INSTANTANEOUS RATE GRAPH 1 GRAPH 2 GRAPH 3 GRAPH 4 D I S P L A C E M E N T D I S P L A C E M E N T D I S P L A C E M E N T D I S P L A C E M E N T

21. Time Using Slopes of Lines in Physics  S  t  t  v Slope of a tangent drawn to a point on a displacement vs time graph gives the instantaneous velocity at that point Slope of a tangent drawn to a point on a velocity vs time graph gives the instantaneous acceleration at that point D I S P L A C E M E N T V E L O C I T Y Time A C C E L E R A T I O N Time

22. Y X X 1 X 2 AREA UNDER THE CURVE FROM X 1 TO X 2 Area =  Y  X (SUM OF THE BOXES) WIDTH OF EACH BOX =  X AREA MISSED - INCREASING THE NUMBER BOXES WILL REDUCE THIS ERROR! Finding Area Under a Curve AS THE NUMBER OF BOXES INCREASES, THE ERROR DECREASES!

24. 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 n Subtract one From the power n - 1 dy/dx is the mathematical Symbol for the derivative Y = k X n dy / dx = n k X n - 1

25. APPLYING THE DERIVATIVE FORMULA GIVEN THE EQUATION FORMAT TO FIND THE DERIVATIVE dy / dx = 5 X 3 3 x 3 - 1 = 2 Using the derivative equation we can find the slope of the y = 5 x 3 equation at any x point. For example, the slope at x = 2 is Slope = 15 x 2 2 = 60. At x = 5, slope = 15 x 5 2 = 375. Derivatives Can be used To find: Velocity, Acceleration, Angular Velocity, Angular Acceleration, Etc. Y = 5 X 3 dy / dx = n k X n - 1 dy / dx = 15 X 2

26. 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 3t 1 and the term 6 is 6t 0. Also, any term to the zero power equals one 2 x -1 1 x -1 0 x -1 dy / dx = 8 t + 3 + 0 = 8 t + 3 y = 4 t + 3 t + 6 2 dy / dt = 4 t + 3 t + 6 2 1 t 0

27. 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 n + 1 DIVIDE THE EQUATION BY THE N + 1 --------------- n + 1 ADD A CONSTANT + C  is the symbol for integration Y = k X n = k X n n + 1 --------------- n + 1 + C  k X dx n d y = 

28. APPLYING THE INTEGRAL FORMULA GIVEN THE EQUATION FORMAT TO FIND THE INTEGRAL 5 X 3 3 + 1 --------------- 3 + 1 + C 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. Y = 5 X 3 = k X n n + 1 --------------- n + 1 + C  k X dx n d y =  = 5 X 4 --------------- 4 , dy = 5 X dX 3 d y = 

29. APPLYING THE INTEGRAL FORMULA Find the area between x = 2 and x = 5 for the equation y = 5X 3 . First find the integral of the equation as shown on the previous frame. The integral was found to be 5/4 X 4 + C. Area 5 2 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) Area - = 761.25 = 5 X 4 --------------- 4 + C = 5 (5) --------------- 4 4 + C 5 (2) 4 4 --------------- + C

31. Physics Mathematics Indicating Direction Up = + Down = - Right = + Left = + y x + + - - Quadrant I Quadrant II Quadrant III Quadrant IV 0 o 90 o 180 o 270 o 360 o Rectangular Coordinates

32. RADIANS = ARC LENGTH / RADIUS LENGTH CIRCUMFERENCE OF A CIRCLE = 2  x RADIUS RADIANS IN A CIRCLE = 2  R / R 1 CIRCLE = 2  RADIANS = 360 O 1 RADIAN = 360 O / 2  = 57.3 O y x + + - - Quadrant I Quadrant II Quadrant III Quadrant IV 0 radians  radians 3/2  radians 2  radians  /2 radians Measuring angles in Radians

33. EXAMPLES OF GEOGRAPHIC DIRECTIONAL MEASUREMENTS NORTH EAST SOUTH WEST NORTHEAST SOUTHWEST SOUTHEAST NORTHWEST NORTH NORTHEAST EAST SOUTHEAST SOUTH SOUTHWEST NOTICE THAT THESE DIRECTIONS ARE NOT PRECISE !

35. A MORE PRECISE METHOD OF GEOGRAPHIC MEASUREMENT EAST NORTH WEST SOUTH 50 0 NORTH OF EAST 25 0 WEST OF SOUTH -45 0 (ANOTHER WAY TO MEASURE ANGLES)

37. The Right Triangle C = the hypotenuse A RIGHT TRIANGLE = the legs Pythagorean Theorem B A & B A + B = C 2 2 2 C = A + B 2 2 A = C - B 2 2 B = C - A 2 2  B C C C A 90 0  90 0    90 0 + + = 180 0

39. Fundamental Trigonometry (SIDE RATIOS)  A B C Sin = A / C  Cos = B / C  Tan  = A / B A C B A B A RIGHT TRIANGLE C

41.  Quadrant III Quadrant IV Quadrant I Quadrant II Sin Cos Tan    + + + + - -  - - +  - + - Trig Function Signs  /2 radians 90 o 0 o 180 o 270 o 360 o y x + + - - 0 radians  radians 3/2  radians 2  radians 

42. Scientific Numbers In science, we often encounter very large and very small numbers. Using scientific numbers makes working with these numbers easier 5,010,000,000,000,000,000,000 a very large number (the number of atoms in a drop of water) 0.000000000000000000000327 a very small number (mass of a gold atom in grams)

43. Scientific Numbers Scientific numbers use powers of 10 3 2 -1 -2 2 523 = 5.23 x 100 = 5.23 x 10 0.0523 = 5.23/100 = 5.23/10 = 5.23 x 10 2 -2 2 100 = 10 x 10 = 10 1000 = 10 x 10 x 10 = 10 0.10 = 1 / 10 = 10 0.01 = 1 / 100 = 1 / 10 = 10 1

44. Scientific Numbers RULE 1 As the decimal is moved to the left The power of 10 increases one value for each decimal place moved 450,000,000 = 450,000,000. x 10 0 Any number to the Zero power = 1 450,000,000 = 450,000,000. x 10 0 4.5 x 10 8 2 3 8 1

45. Scientific Numbers RULE 2 As the decimal is moved to the right The power of 10 decreases one value for each decimal place moved 0.0000072 = 0.0000072 x 10 0 Any number to the Zero power = 1 0 7.2 x 10 -6 -2 -3 -6 -1 0.0000072 = 0.0000072 x 10

46. Scientific Numbers RULE 3 When scientific numbers are multiplied The powers of 10 are added 2 3 5 (2 + 3) 2 3 100 x 1000 = 100,000 100 = 10 1000 = 10 10 x 10 = 10 = 10 = 100,000 (3 x 10 ) x ( 2 x 10 ) = 6 x 10 2 3 5

47. Scientific Numbers RULE 4 When scientific numbers are divided The powers of 10 are subtracted 4 2 2 (4 - 2) 4 2 10000 / 100 = 100 10000 = 10 100 = 10 10 / 10 = 10 = 10 = 100 (5 x 10 ) / (2 x 10 ) = 2.5 x 10 4 2 2

48. Scientific Numbers RULE 5 When scientific numbers are raised to powers The powers of 10 are multiplied 4 2 (100) = 10,000 100 = 10 2 (10 ) = 10 = 10 = 10,000 2 2 (2 x 2) (3000) = 9,000,000 (3 x 10 ) = 3 x 10 = 9 x 10 2 2 2 3 (2 x 3) 6

49. Scientific Numbers RULE 6 Roots of scientific numbers are treated as fractional powers. The powers of 10 are multiplied square root = 1/2 power cube root = 1/3 power 10,000 = (10,000) 1/2 10,000 = 10 4 (10,000) = (10 ) = 10 = 100 1/2 4 1/2 (1/2 x 4) (9 x 10 ) = 9 x (10 ) = 3 x 10 6 1/2 1/2 1/2 6 3

50. Scientific Numbers RULE 7 When scientific numbers are added or subtracted The powers of 10 must be the same for each term. 2.34 x 10 + 4.24 x 10 2 3 --------- Powers of 10 are Different. Values Cannot be added ! 2.34 x 10 = 0.234 x 10 3 0.234 x 10 + 4.24 x 10 3 --------- 3 Power are now the Same and values Can be added. 4.47 x 10 3 2 Move the decimal And change the power Of 10

53. THE END