Fundamentals of Engineering (FE) Exam Mathematics Review

Transcription

1 Fundamentals of Engineering (FE) Exam Mathematics Review Dr. Garey Fox Professor and Buchanan Endowed Chair Biosystems and Agricultural Engineering October 16, 2014 Reference Material from FE Review Instructor s Manual, Prepared by Gregg C. Wagener, PE, Professional Publications, Inc

2 Straight Line General form of straight line: Ax + By + C = 0 NOTE: Highest exponent for any variable is 1

3 Straight Line Standard form (slope-intercept form): y = mx + b Point-slope form: y-y 1 =m(x-x 1 ) Equation for the slope: m = (y 2 -y 1 )/(x 2 -x 1 ) Distance between two points: d 2 y y x x1

4 Intersecting Straight Lines Angle between the lines: tan 1 m 1 2 m m m If lines are perpendicular: m 1 1 m 2

5 Example Problem What is the slope of the line 2y = 2x + 4? (A) 1 (B) 1/2 (C) -2 (D) Infinite

6 Example Problem

7 Algebra Solving two linear equations simultaneously First, look for a simple substitution Second, look for a simple reduction

8 Quadratic Equation Any equation of the form... ax 2 bx c 0 Roots of the equation (x 1,x 2 ): x 1, x 2 b 2 b 2a 4ac

9 Quadratic Equation Discriminant determines the roots... b 2 4ac 0 Real and Unequal b 2 4ac 0 Real and Equal b 2 4ac 0 Complex and Unequal

10 Example

11 Cubic Equation Any equation of the form... ax 3 bx 2 cx d 0 Roots of the equation (x 1,x 2,x 3 ): Simplify and find easiest roots Look for answers that can be eliminated Plug and chug!

12 Example

13 Any of several curves produced by passing a plane through a cone Conic Sections

14 Conic Sections Two Angles: q = angle between the vertical axis and the cutting plane f = cone-generating angle Eccentricity, e, of a conic section: e cosq cosf

15 Conic Sections Quadratic Equation: Ax 2 2Bxy Cy If A = C = 0, then conic section = line If A = C 0, then conic section = circle If A C: B 2 -AC<0, then conic section = ellipse B 2 -AC>0, then conic section = hyperbola B 2 -AC=0, then conic section = parabola 2 2Dx 2Ey F 0

16 Example Ax 2 2Bxy Cy 2 2Dx 2Ey F 0

17 Parabola

18 Parabola For Center (vertex) at (h,k), focus at (h+p/2, k), directrix at x=h-p/2 and that opens horizontally 2 ( y k) 2p( x h) Opens to Right if p>0 Opens to Left if p<0

19 Example

20 Ellipse

21 Ellipse For Center (vertex) at (h,k), semimajor distance (a) and semiminor distance (b) ( x h) 2 a 2 ( y k) 2 b 2 1 e c / a

22 Circle Special ellipse ( x h) ( y k) r

23 Circle Length, t, to a circle from a point (x, y ): t ( x h) ( y k) r

24 Example

25 Hyperbola

26 Hyperbola For Center (vertex) at (h,k) and opening horizontally ( x h) 2 a 2 ( y k) 2 b 2 1 e c / a, e 1

27 Three-Dimensional Objects Sphere centered at (h,k,m) with radius r: Distance between two points in 3-d space: ) ( ) ( ) ( r m z k y h x ) ( ) ( ) ( z z y y x x d

28 Mensuration Mensuration (measurements) of perimeter, area, and other geometric properties Handbook for Formulas!

29 Example

30 Logarithms Think of logarithms as exponents... b c x Exponent is c and expression above is the logarithm of x to the base b log b ( x) c b Base for common logs is 10 (log=log 10 ) Base for natural logs is e (ln=log e ), e = Identities - HANDBOOK! c x

31 Trigonometry sin, cos, tan cot = 1/tan, csc = 1/sin, sec=1/cos Law of sines and cosines! Identities - HANDBOOK!

32 Trigonometry Plug in sin and cos for all tan, cot, csc, and sec Simplify and look for a simple identity OR work backwards by simplifying the possible answers

33 Example

34 Complex Numbers Combination of real and imaginary numbers (square root of a negative number) i 1 Rectangular Form: a ib

35 Complex Numbers Identities - HANDBOOK! Algebra is done separately for real and imaginary parts! Multiplying: Rectangular Form: Note that i 2 =-1 Polar Coordinates: Converting z = a + ib to z = r(cos q +i sin q) HANDBOOK! Multiplication: Magnitude multiply/divide, Phase angle add/subtract

36 Complex Numbers Another notation for polar coordinates: z = re iq (Euler s Identity HANDBOOK!) Convert Rectangular/Polar - HANDBOOK! Roots - the kth root, w, of a complex number z = r(cos q +i sin q) is given by: o k q 360 q 360 w r cos n isin n k k k k o

37 Example

38 Example

39 Matrices m x n = number of rows x number of columns Square Matrix: m=n (order) Multiplication: Two matrices: A = m x n B = n x s AB = m x s BA = Not Possible

40 Matrices Multiplication Addition: only possible if matrices have same number of rows and columns

41 Identity Matrix: Matrices Transpose of a m x n matrix is n x m matrix constructed by taking i th row and making it the i th column

42 Matrices Determinants: Formulas in HANDBOOK! Minor of element a i,j = determinant when row i and j are crossed out (if i+j is even, then multiply the determinant by 1 and if odd, then multiply the determinant by -1)

43 Matrices Cofactor Matrix = minor for all elements of the original matrix with appropriate sign cofactor of 1 is Classical Adjoint = transpose of the cofactor matrix, adj(a)

44 Matrices Inverse = classical adjoint matrix divided by the determinant (HANDBOOK!)

45 Vectors Scalar, Vector, Tensor Unit Vectors (i, j, k) Vector Operations - Clearly outlined in HANDBOOK! Dot Product, Cross Product Gradient, divergence, and curl (pg. 24) k b j b i b B k a j a i a A z y x z y x ˆ ˆ ˆ ˆ ˆ ˆ

46 Example

47 Example

48 Example For the three vectors A, B and C, what is the product A (BC)? A = 6i + 8j + 10k B = i + 2j + 3k C = 3i + 4j + 5k (A) 0 (B) 64 (C) 80 (D) 216

49 Derivatives: Differential Calculus Definition of a Derivative: Relations among Derivatives (not in handbook): Tables of Derivatives

50 Differential Calculus Slope (A) 128 (B) 64 (C) 9 (D) 4 (A) 0.25 (B) 0.5 (C) 0.75 (D) 1.0

51 Maxima and Minima- Differential Calculus

52 Inflection Points: Differential Calculus

53 Partial Derivatives: Differential Calculus

54 Differential Calculus Curvature (K) of any Curve at P: Rate of change of inclination with respect to its arc length Radius of Curvature (R) Radius of a circle that would be tangent to a function at any point

55 Differential Calculus

56 Limits - Differential Calculus

57 Differential Calculus

58 Differential Calculus

59 Constant of Integration - Integral Calculus

60 Indefinite Integrals - Integral Calculus

61 Integral Calculus

62 Integral Calculus

63 Integral Calculus

64 Integral Calculus

65 Definite Integrals - Integral Calculus

66 Integral Calculus Average Value - (A) 1 (B) 2 (C) 4 (D) 8

67 Areas - Integral Calculus

68 Integral Calculus (A) 13/35 (B) 11/12 (C) 41/32 (D) 2

69 Differential Equations Order of DE highest order derivative First-order Homogeneous Equations:

70 Differential Equations

71 Differential Equations Separable Equations:

72 Differential Equations Second-Order Homogeneous Equations:

73 Differential Equations

74 Example

75 Combinations: Probability and Statistics

76 Permutations: Probability and Statistics

77 Probability and Statistics Laws of Probability:

78 Probability and Statistics Joint Probability:

79 Probability and Statistics

80 Probability and Statistics

81 Probability and Statistics Probability Functions: Binomial Distribution:

82 Probability and Statistics

83 Probability and Statistics Probability Density Functions:

84 Probability and Statistics Statistical Treatment of Data: Arithmetic Mean: Weighted Arithmetic Mean:

85 Probability and Statistics Statistical Treatment of Data: Median:

86 Probability and Statistics Statistical Treatment of Data: Mode: Variance: Standard Deviation:

87 Probability and Statistics Normal Distribution (Gaussian): averages of n observations tend to become normally distributed as n increases

88 Probability and Statistics Normal Distribution (Gaussian): when mean is zero and standard deviation is 1.0 called standardized or unit normal distribution: Unit Normal Distribution Table

89 Probability and Statistics Convert distribution to unit normal distribution:

90 Probability and Statistics