Mathematical Fundamentals (Part 1)

Transcription

1 Mathematical Fundamentals (Part 1)

2 Mathematics Chapter 1

3 ALGEBRAIC OPERATIONS (+)(+) = POSITIVE (-)(-) = POSITIVE (+)(-) = NEGATIVE (-)(+) = NEGATIVE

4 a x b a x (-b) = ab = -ab a(b + c) = ab + ac a(b - c) = ab ac (a + b)c = ac + bc a + b = b + a a b b a

5 EXPONENTS 10 4 = 10 x 10 x 10 x = 10 x = = 1 0 = 1 a 0 = 1 x 0 = 1 (3 x 10) 0 = 1 3 x 10 0 = 3

6 MULTIPLICATION 10 x 10 3 = x 10 7 = = x 10 = = 10 a 3 x a 5 = a 8 a x x a y = a x+y

7 - x -3 = -5 1 = 5 1 = 3 (b 3 ) = (b 3 ) (b 3 ) = 4b 6 (a n )(a m ) = a n+m RULE: Like Bases Are Added

8 DIVISION a a 5 3 a a = 10-5 = 10-3

9 = = 3 - = 3 1 = a n a m = a n-m RULE: Like Bases Are Subtracted

10 RADICALS

11 ( 3 8)

12 CHANGING POWERS a b 3 3 b a 3 3

13 , ( ) ( 310 ) ( 10 ) 3 3 ( 9 10 ) ( )

14 Linear Equation 1 Unknown ax + b = x 3x x 3 36x 10 8x 7 x 1 4

15 Linear Equation - Unknowns x 3y 1 3x 5y 7 By Substitution: 3y 1 x y 1 3 x 3

16 x 4 Subst. X=-4 in equation #1 1 x 3x x 3x 7 3 b g 3 9x 510x 81 3 b g y y 1 3y 9 y 3 b gb g 1, 1, b gb g 4, 3 4, 3

17 Quadratic Formula Solve 5x - 8x = -3 using the quadratic formula. We first find standard form and determine a, b, and c: 5x - 8x + 3 = 0, a = 5, b = -8, c = 3 We then use the quadratic formula: x b b ac 4 a

18 x b g b g x x x 8 10 x 1 or x 3 5

19 5x 8x 3 0 ( 5x 3)( x 1) 0 5x 3 0 or x 1 0 5x 3 or x 1 3 x or x 1 5

20 RATIO It is required to find how many feet of rolled thread stock would be needed to produce 1500 knurls if 50 of these knurls can be made from 5 feet of material. 50:5 = 1500:x 50x = 37,500 x = 750 ft.

21 Assume a job shop produces a given number of stampings in 7 hours per week with a work force of 300 people. Work hours are to be cut back to 54. How many additional people will be needed to maintain production? Note that the number of people and the time involved vary inversely with each other. As the number of people increase, the time decreases. Thus: 300:x=54:7 54x=1600 x=400 people required 100 additional people required

22 PERCENTAGE By purchasing a new piece of equipment, 10 pieces can be made in the same length of time normally used for 90. Express in percent the gain in production from the new piece of equipment over the previous method used % 90

23 An alloy contains 37 pounds of copper and 15.5 pounds of nickel. What percent of the alloy is copper? = lbs = 96%

24 AREA h b A = bh

25 h b A 1 bh

26 a h b 1 A a b h ( )

27 A D r 4

28 VOLUME A r h

29 VOLUME A h( R r )

30 SLOPE OF A LINE slope rise run 3 4 Intercepts: X axis: 4 Y axis: -3

31 SLOPE INTERCEPT FORM Given: y = mx + b m = slope b = y intercept 3x-4y = 1-4y = 1-3x 4y = -1+3x y = 3/4x-3

32 INTERCEPT/INTERCEPT FORM x A B y x y x-4y = 1-4y = 1-3x 4y = -1+3x y = 3/4x-33 Given: A= 4, B= -3

33 Given points: Point 1: (3,-7) Point : (-4,5) y y1 y y1 x x x x ( 7) y ( 7) 4 3 x 3 1 y 7 7 x 3 1( x 3) 7( y 7) 1x 36 7 y 49 1x13 7 y 7y 1x y x 7 7

34 .10 A RIGHT TRIANGLES Sol C A.10 C A B ( 35. ) C A B( 10. ) B ( 35. ) ( 10. ) B B B B B B Solving Right Triangles Solving Right Triangles 1B B B. 8 B.10 A.10 A C 3.50 C C 3.50 B B 3.50 C A B B B B ( 35. ) ( 10. ) B B B. 8 C A B ( 35. ) ( 10. ).10 A B

35 sine = cosine tangent cosecant Trigonometry = = side opposite hypotenuse side adjacent hypotenuse side opposite side adjacent = hypotenuse side opposite a c b c a b c a

36 sin cos tan csc ordinate radius abscissa radius ordinate abscissa radius ordinate y r x r y x r x

37 LAW OF SINES The Law of Sines b C a A c B a b c sin A sin B sinc

38 Given: A = 45 a = 55 B = 30 b =? C =? c =? C = ( ) = sin 45 b sin b b b sin 45 c sin c c 5. 8 c

39 Probability and Reliability The probability of A or B occurring is: P(A or B) = P(A) +P(B) P (A and B) Mutually exclusive events if two events cannot occur simultaneously For example, in a coin tossing experiment if a head occurs then a tail cannot. P(A and B) = 0 P(A or B) = P(A) + P(B)

40 Probability and Reliability Independent events if two events can occur in a single experimental trial however one event does not affect the probability of the occurrence of the other For example when tossing a pair of dice, rolling a four on the first die and a four on the second die are independent events. P(A and B) = P(A) P(B)

41 Measure of Central Tendency Measure of central tendency is a numeric value that describes the central position or location of the data mean median mode

42 Mean n i1 X N i true mean N totalnumber of observations in the population X i individual observations X n i1 X n i X sample mean n totalnumber of observations in the sample X i individual observations

43 Median The median is the middle observation in a group of data ordered by magnitude. The data are ordered in ascending or descending order and counted. The median is halfway through this ordered list. If there are an even number of observations, the median is the average of the two in the middle of the ordered list.

44 Mode The mode is the value that occurs most frequently.

45 Measure of Variation Range Range (R) = Maximum x i - Minimum x i Variance measure of the variability in data. Standard Deviation more practical to use than variance since the units of standard deviation usually match the units in the problem such as inches or pounds interpretations about the variability are more easily drawn with standard deviations

46 Population Variance N N ( xi ) xi N i1 i1 N N Where, = the population mean N = population measurements x i = each individual data point

47 Sample Variance s n n ( xi x) xi n( x) i1 i1 n1 n1 x Where, = the sample mean x i = each individual data point n = sample size

48 Population Standard Deviation N N ( xi ) xi N i 1 i 1 N N

49 Sample Standard Deviation s n n ( xi x) xi n( x) i1 i1 s n1 n1

50 Central Limit Theorem The Central Limit Theorem states that the distributions of sample means from an infinite population will approach a normal distribution as the sample size increases. The size of the sample that will give a nearly normal distribution depends on how non-normal the population is.

51 Normal Distribution

52 Normal Distribution 68.6% of the observations fall between 95.46% of the observations fall between 99.73% of the observations fall between u x u u x 3 x

53 Standard Normal Distribution Because the mean and standard deviation of a normal distribution can take on many different values depending on the situation it is convenient to work with a standard normal distribution where z is a standardized normal random variable based on any normal x distribution with mean u and standard deviation z ( X ) x

54 Standard Normal Distribution Using a table which provides areas under a normal curve the percentage of observations above or below a certain value can be calculated.

55 Example Problem If the diameter of shafts is normally distributed with a mean of 1.00 and a standard deviation of 0.01 what is the probability that a given shaft will have a diameter between and 1.005? z z From a cumulative area under the normal curve table, the probability will be A1- A = = or 6.47%.

56 3 s t 4t 3t ds 3t 8t 3 v dt dv3 6t 8 a ss tt3 dt 4 t 3t ds ds 3t 8t 3 v dt t dt dv dv 6t 8 a dt 6t dt CALCULUS 3t 8t 3 0 ( 3t 1)( t 3) 0 t 3 set set velocity = 0= 0 1 3t3 t8 t8 t ( 3( t3 t 1 )( 1)( t t 3 ) 3 ) 0 0 t t 3 3 When t = 3, a = 1 t 1 t 3 t = 1 3 a = 3, When t = 3, a = +10 When t = 3, a = +10 t = 1 t = 1 a = 10 3, a = 10 3,

57 CALCULUS 6t 8 6t 8t c 1 3t 8t c 3 8 t t c 3 3t 8t 3 3 t 4t c c