Fundamentals of Engineering (FE) Exam Mathematics Review

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-y1=m(x-x1) • Equation for the slope: m = (y2-y1)/(x2-x1) • 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 (x1,x2):

9. Quadratic Equation • Discriminant determines the roots...

10. Example

11. Cubic Equation • Any equation of the form... • Roots of the equation (x1,x2,x3): • 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: q = angle between the vertical axis and the cutting plane f = 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 ¹ 0, then conic section = circle • If A ¹ C: • B2-AC<0, then conic section = ellipse • B2-AC>0, then conic section = hyperbola • B2-AC=0, then conic section = parabola

16. Example

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 • 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)

22. Circle • Special ellipse

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

24. Example

25. Hyperbola

26. Hyperbola • 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=log10) • Base for natural logs is e (ln=loge), 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 i2=-1 • Polar Coordinates: Converting z = a + ib to z = r(cosq +i sin q) • HANDBOOK! • Multiplication: Magnitude multiply/divide, Phase angle add/subtract

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

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. Matrices • Identity Matrix: • Transpose of a m x n matrix is n x m matrix constructed by taking ith row and making it the ith column

42. Matrices • Determinants: Formulas in HANDBOOK! • Minor of element ai,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

47. Example

48. Example • 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