1. Fundamentals of Engineering (FE) Exam Mathematics Review Dr. Garey Fox Associate Professor and Buchanan Chair Biosystems and Agricultural Engineering March 2013 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 • Focus (point F) • Directrix • Symmetric about Parabolic Axis • Eccentricity, e = 1

19. 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

20. Example

21. Ellipse

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

23. Circle • Special ellipse

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

25. Example

26. Hyperbola

27. Hyperbola • For Center (vertex) at (h,k) and opening horizontally

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

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

30. Example

31. 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!

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

33. 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

34. Example

35. Complex Numbers • Combination of real and imaginary numbers (square root of a negative number) • Rectangular Form:

36. 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

37. 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:

38. Example

39. Example

40. 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

41. Matrices • Multiplication • Addition: only possible if matrices have same number of rows and columns

42. 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

43. 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)

44. 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)

45. Matrices • Inverse = classical adjoint matrix divided by the determinant (HANDBOOK!)

46. 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)

47. Example

48. Example

49. 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

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