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:

4. Intersecting Straight Lines Angle between the lines: If lines are perpendicular:

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... Roots of the equation (x 1,x 2 ):

9. Quadratic Equation Discriminant determines the roots...

10. Example

11. Cubic Equation Any equation of the form... 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. Conic Sections Any of several curves produced by passing a plane through a cone

14. Conic Sections Two Angles:  = angle between the vertical axis and the cutting plane  = cone-generating angle Eccentricity, e, of a conic section:

15. Conic Sections Quadratic Equation: 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

16. Example

17. Parabola

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

19. Example

20. Ellipse

21. For Center (vertex) at (h,k), semimajor distance (a) and semiminor distance (b)

22. Circle Special ellipse

23. Circle Length, t, to a circle from a point (x’, y’):

24. Example

25. Hyperbola

26. For Center (vertex) at (h,k) and opening horizontally

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

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

29. Example

30. Logarithms Think of logarithms as exponents... – Exponent is c and expression above is the logarithm of x to the base b – Base for common logs is 10 (log=log 10 ) – Base for natural logs is e (ln=log e ), e = 2.71828 Identities - HANDBOOK!

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) Rectangular Form:

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  +i sin  ) – HANDBOOK! – Multiplication: Magnitude multiply/divide, Phase angle add/subtract

36. Complex Numbers Another notation for polar coordinates: z = re i  (Euler’s Identity…HANDBOOK!) Convert Rectangular/Polar - HANDBOOK! Roots - the kth root, w, of a complex number z = r(cos  +i sin  ) is given by:

37. 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. Matrices Identity Matrix: 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)

46. Example

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

49. Differential Calculus Derivatives: – 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. Differential Calculus Maxima and Minima-

52. Differential Calculus Inflection Points:

53. Differential Calculus Partial Derivatives:

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 -

57. Differential Calculus

59. Integral Calculus Constant of Integration -

60. Integral Calculus Indefinite Integrals -

61. Integral Calculus

65. Definite Integrals -

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

67. Integral Calculus Areas -

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. Separable Equations:

72. Differential Equations Second-Order Homogeneous Equations:

73. Differential Equations

74. Example

75. Probability and Statistics Combinations:

76. Probability and Statistics Permutations:

77. Probability and Statistics Laws of Probability:

78. Probability and Statistics Joint Probability:

79. Probability and Statistics

81. Probability Functions: Binomial Distribution:

82. Probability and Statistics

83. 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. Convert distribution to unit normal distribution: Probability and Statistics