1. FE Mathematics Review Dr. Scott Molitor Associate Professor
Undergraduate Program Director
Department of Bioengineering

2. Topics covered Analytic geometry Integral calculus
Equations of lines and curves
Distance, area and volume
Trigonometric identities
Integral calculus
Integrals and applications
Numerical methods
Differential equations
Solution and applications
Laplace transforms
Difference equations and Z transforms
Differential calculus
Derivatives and applications
Limits and L’Hopital’s rule
Algebra
Complex numbers
Matrix arithmetic and determinants
Vector arithmetic and applications
Progressions and series
Numerical methods for finding solutions of nonlinear equations
Probability and statistics
Rules of probability
Combinations and permutations
Statistical measures (mean, S.D., etc.)
Probability density and distribution functions
Confidence intervals
Hypothesis testing
Linear regression

3. Tips for taking exam Use the reference handbook
Know what it contains
Know what types of problems you can use it for
Know how to use it to solve problems
Refer to it frequently
Work backwards when possible
FE exam is multiple choice with single correct answer
Plug answers into problem when it is convenient to do so
Try to work backwards to confirm your solution as often as possible
Progress from easiest to hardest problem
Same number of points per problem
Calculator tips
Check the NCEES website to confirm your model is allowed
Avoid using it to save time!
Many answers do not require a calculator (fractions vs. decimals)

4. Equations of lines
What is the general form of the equation for a line whose x-intercept is 4 and y-intercept is -6?
(A) 2x – 3y – 18 = 0
(B) 2x + 3y + 18 = 0
(C) 3x – 2y – 12 = 0
(D) 3x + 2y + 12 = 0

5. Equations of lines
What is the general form of the equation for a line whose x-intercept is 4 and y-intercept is -6?
(A) 2x – 3y – 18 = 0
(B) 2x + 3y + 18 = 0
(C) 3x – 2y – 12 = 0
(D) 3x + 2y + 12 = 0
Try using standard form
Handbook pg 3: y = mx + b
Given (x1, y1) = (4, 0)
Given (x2, y2) = (0, -6)
Answer is (C)

6. Equations of lines
What is the general form of the equation for a line whose x-intercept is 4 and y-intercept is -6?
(A) 2x – 3y – 18 = 0
(B) 2x + 3y + 18 = 0
(C) 3x – 2y – 12 = 0
(D) 3x + 2y + 12 = 0
Work backwards
Substitute (x1, y1) = (4, 0)
Substitute (x2, y2) = (0, -6)
See what works
Answer is (C)

7. Trigonometry For some angle q, csc q = -8/5. What is cos 2q? (A) 7/32
(B) 1/4
(C) 3/8
(D) 5/8

8. Trigonometry For some angle q, csc q = -8/5. What is cos 2q?
(B) 1/4
(C) 3/8
(D) 5/8
Use trigonometric identities on handbook page 5
Answer is (A)
Confirm with calculator
First find q = csc-1(-8/5)
Then find cos 2q

9. Polar coordinates
What is rectangular form of the polar equation r2 = 1 – tan2 q?
(A) –x2 + x4y2 + y2 = 0
(B) x2 + x2y2 - y2 - y4 = 0
(C) –x4 + y2 = 0
(D) x4 – x2 + x2y2 + y2 = 0

10. Polar coordinates
What is rectangular form of the polar equation r2 = 1 – tan2 q?
(A) –x2 + x4y2 + y2 = 0
(B) x2 + x2y2 - y2 - y4 = 0
(C) –x4 + y2 = 0
(D) x4 – x2 + x2y2 + y2 = 0
Polar coordinate identities on handbook page 5
Answer is (D)

11. Matrix identities
For three matrices A, B and C, which of the following statements is not necessarily true?
(A) A + (B + C) = (A + B) + C
(B) A(B + C) = AB + AC
(C) (B + C)A = AB + AC
(D) A + (B + C) = C + (A + B)

12. Matrix identities
For three matrices A, B and C, which of the following statements is not necessarily true?
(A) A + (B + C) = (A + B) + C
(B) A(B + C) = AB + AC
(C) (B + C)A = AB + AC
(D) A + (B + C) = C + (A + B)
Matrix identities on handbook page 7
Answer is (C)
Should know (A) and (D) are true from linear algebra
Answer (B) appears as matrix identity in handbook page 7
Therefore can eliminate (C) as being true

13. Vector calculations
For three vectors A = 6i + 8j + 10k B = i + 2j + 3k C = 3i + 4j + 5k, what is the product A·(B x C)?
(A) 0
(B) 64
(C) 80
(D) 216

14. Vector calculations
For three vectors A = 6i + 8j + 10k B = i + 2j + 3k C = 3i + 4j + 5k, what is the product A·(B x C)?
(A) 0
(B) 64
(C) 80
(D) 216
Vector products on handbook page 6
Answer is (A)

15. Vector calculations
For three vectors A = 6i + 8j + 10k B = i + 2j + 3k C = 3i + 4j + 5k, what is the product A·(B x C)?
(A) 0
(B) 64
(C) 80
(D) 216
Vector products on handbook page 6
Answer is (A)
Aside: why is the answer zero? A dot product is only zero when two vectors A and (B x C) are perpendicular. But this is the case! A and C are parallel (A = 2C), and (B x C) is perpendicular to C, hence perpendicular to A!

16. Geometric progression
The 2nd and 6th terms of a geometric progression are 3/10 and 243/160. What is the first term of the sequence?
(A) 1/10
(B) 1/5
(C) 3/5
(D) 3/2

17. Geometric progression
The 2nd and 6th terms of a geometric progression are 3/10 and 243/160. What is the first term of the sequence?
(A) 1/10
(B) 1/5
(C) 3/5
(D) 3/2
Geometric progression on handbook page 7
Answer is (B)

18. Geometric progression
The 2nd and 6th terms of a geometric progression are 3/10 and 243/160. What is the first term of the sequence?
(A) 1/10
(B) 1/5
(C) 3/5
(D) 3/2
Geometric progression on handbook page 7
Answer is (B)
Confirm answer by calculating l2 and l6 with a = 1/5 and r = 3/2.

19. Roots of nonlinear equations
Newton’s method is being used to find the roots of the equation f(x) = (x – 2)2 – 1. Find the 3rd approximation if the 1st approximation of the root is 9.33
(A) 1.0
(B) 2.0
(C) 3.0
(D) 4.0

20. Roots of nonlinear equations
Newton’s method is being used to find the roots of the equation f(x) = (x – 2)2 – 1. Find the 3rd approximation if the 1st approximation of the root is 9.33
(A) 1.0
(B) 2.0
(C) 3.0
(D) 4.0
Newton’s method on handbook page 13
Answer is (D)

21. Limits What is the limit of (1 – e3x) / 4x as x  0? (A) -∞ (B) -3/4
(C) 0
(D) 1/4

22. Limits What is the limit of (1 – e3x) / 4x as x  0?
(B) -3/4
(C) 0
(D) 1/4
L’Hopital’s rule on handbook page 8
Answer is (B)

23. Limits What is the limit of (1 – e3x) / 4x as x  0?
(B) -3/4
(C) 0
(D) 1/4
L’Hopital’s rule on handbook page 8
Answer is (B)
You should apply L’Hopital’s rule iteratively until you find limit of f(x) / g(x) that does not equal 0 / 0.
You can also use your calculator to confirm the answer, substitute a small value of x = 0.01 or

24. Application of derivatives
The radius of a snowball rolling down a hill is increasing at a rate of 20 cm / min. How fast is its volume increasing when the diameter is 1 m?
(A) m3 / min
(B) 0.52 m3 / min
(C) 0.63 m3 / min
(D) 0.84 m3 / min

25. Application of derivatives
The radius of a snowball rolling down a hill is increasing at a rate of 20 cm / min. How fast is its volume increasing when the diameter is 1 m?
(A) m3 / min
(B) 0.52 m3 / min
(C) 0.63 m3 / min
(D) 0.84 m3 / min
Derivatives on handbook page 9; volume of sphere on handbook page 10
Answer is (C)

26. Application of derivatives
The radius of a snowball rolling down a hill is increasing at a rate of 20 cm / min. How fast is its volume increasing when the diameter is 1 m?
(A) m3 / min
(B) 0.52 m3 / min
(C) 0.63 m3 / min
(D) 0.84 m3 / min
Derivatives on handbook page 9; volume of sphere on handbook page 10
Answer is (C)
Convert cm to m, convert diameter to radius, and confirm final units are correct.

27. Evaluating integrals
Evaluate the indefinite integral of f(x) = cos2x sin x
(A) -2/3 sin3x + C
(B) -1/3 cos3x + C
(C) 1/3 sin3x + C
(D) 1/2 sin2x cos2x + C

28. Evaluating integrals
Evaluate the indefinite integral of f(x) = cos2x sin x
(A) -2/3 sin3x + C
(B) -1/3 cos3x + C
(C) 1/3 sin3x + C
(D) 1/2 sin2x cos2x + C
Apply integration by parts on handbook page 9
Answer is (B)

29. Evaluating integrals
Evaluate the indefinite integral of f(x) = cos2x sin x
(A) -2/3 sin3x + C
(B) -1/3 cos3x + C
(C) 1/3 sin3x + C
(D) 1/2 sin2x cos2x + C
Alternative method is to differentiate answers
Answer is (B)

30. Applications of integrals
What is the area of the curve bounded by the curve f(x) = sin x and the x-axis on the interval [p/2, 2p]?
(A) 1
(B) 2
(C) 3
(D) 4

31. Applications of integrals
What is the area of the curve bounded by the curve f(x) = sin x and the x-axis on the interval [p/2, 2p]?
(A) 1
(B) 2
(C) 3
(D) 4
Need absolute value because sin x is negative over interval [p, 2p]
Answer is (C)

32. Differential equations
What is the general solution to the differential equation y’’ – 8y’ + 16y = 0?
(A) y = C1e4x
(B) y = (C1 + C2x)e4x
(C) y = C1e-4x + C1e4x
(D) y = C1e2x + C2e4x

33. Differential equations
What is the general solution to the differential equation y’’ – 8y’ + 16y = 0?
(A) y = C1e4x
(B) y = (C1 + C2x)e4x
(C) y = C1e-4x + C1e4x
(D) y = C1e2x + C2e4x
Solving 2nd order differential eqns on handbook page 12
Answer is (B)

34. Differential equations
What is the general solution to the differential equation y’’ – 8y’ + 16y = 0?
(A) y = C1e4x
(B) y = (C1 + C2x)e4x
(C) y = C1e-4x + C1e4x
(D) y = C1e2x + C2e4x
Solving 2nd order differential eqns on handbook page 12
Answer is (B)
In this case, working backwards could give an incorrect answer because answer (A) would also work. The answer in (B) is the sum of two terms that would satisfy the differential equation, one of these terms is the same as answer (A).

35. Laplace transforms
Find the Laplace transform of the equation f”(t) + f(t) = sin bt where f(0) and f’(0) = 0
(A) F(s) = b / [(1 + s2)(s2 + b2)]
(B) F(s) = b / [(1 + s2)(s2 - b2)]
(C) F(s) = b / [(1 - s2)(s2 + b2)]
(D) F(s) = s / [(1 - s2)(s2 + b2)]

36. Laplace transforms
Find the Laplace transform of the equation f”(t) + f(t) = sin bt where f(0) and f’(0) = 0
(A) F(s) = b / [(1 + s2)(s2 + b2)]
(B) F(s) = b / [(1 + s2)(s2 - b2)]
(C) F(s) = b / [(1 - s2)(s2 + b2)]
(D) F(s) = s / [(1 - s2)(s2 + b2)]
Laplace transforms on handbook page 174 (EECS section)
Answer is (A)

37. Probability of an outcome
A marksman can hit a bull’s-eye 3 out of 4 shots. What is the probability he will hit a bull’s-eye with at least 1 of his next 3 shots?
(A) 3/4
(B) 15/16
(C) 31/32
(D) 63/64

38. Probability of an outcome
A marksman can hit a bull’s-eye 3 out of 4 shots. What is the probability he will hit a bull’s-eye with at least 1 of his next 3 shots?
(A) 3/4
(B) 15/16
(C) 31/32
(D) 63/64
Answer is (D)
Let H = hit, M = miss, Prob(H) = ¾, Prob(M) = ¼
Use combinations for next three shots
Find Prob(HMM + MHM + MMH + HHM + ...)
Easier method: Prob(at least one hit) = 1 – Prob(no hits)
1 – Prob(no hits) = 1 – Prob(MMM)
Prob(MMM) = Prob(M)3 = (1/4)3 = 1/64
Answer is 1 – 1/64 = 63/64

39. Normal distribution
Exam scores are distributed normally with a mean of 73 and a standard deviation of 11. What is the probability of finding a score between 65 and 80?
(A)
(B)
(C)
(D)

40. Normal distribution
Exam scores are distributed normally with a mean of 73 and a standard deviation of 11. What is the probability of finding a score between 65 and 80?
(A)
(B)
(C)
(D)
Standard normal tables on handbook page 20
Answer is (B)
Let X = a random score, find Prob(65 < X < 80)
X is normally distributed with mean 72 and S.D. 11
(65 – 72) / 11 = ≈ -0.7
(80 – 72) / 11 = 0.64 ≈ 0.6
Prob(65 < X < 80) ≈ Prob(-0.7 < Z < 0.6)
Convert Prob(-0.7 < Z < 0.6)
Prob(Z < 0.6) – Prob(Z < -0.7)
Prob(Z < 0.6) – Prob(Z > 0.7)
F(0.6) – R(0.7) from table
Prob(65 < X < 80) ≈ – =

41. Confidence intervals
What is the 95% confidence interval for the mean exam score if the mean is 73 and the standard deviation is 11 from 25 scores?
(A) 73 ± 4.54
(B) 73 ± 0.91
(C) 73 ± 4.31
(D) 73 ± 0.86

42. Confidence intervals
What is the 95% confidence interval for the mean exam score if the mean is 73 and the standard deviation is 11 from 25 scores?
(A) 73 ± 4.54
(B) 73 ± 0.91
(C) 73 ± 4.31
(D) 73 ± 0.86
Confidence intervals on handbook page 19
ta values handbook page 21
Answer is (A)
Use formula for population standard deviation unknown
Formula is
Look up ta/2, n
a = 1 – 0.95 = 0.05
a/2 = 0.025
n = 25 – 1 = 24 degrees of freedom
t0.025, 24 = on page 21
Calculate confidence interval 73 ± (2.064) (11) / √25
Answer is 73 ± 4.54

43. Hypothesis testing
You sample two lots of light bulbs for mean lifetime. The first lot mean = 792 hours, S.D. = 35 hours, n = 25. The second lot mean = 776 hours, S.D. = 24 hours, n = 20. Determine with 95% confidence whether light bulbs from the first lot last longer than those from the second lot. Provide a statistic value.
(A) First lot lasts longer, t0 = -1.96
(B) No difference, z0 = 1.81
(C) No difference, t0 = 1.74
(D) First lot lasts longer, t0 = 1.96

44. Hypothesis testing
You sample two lots of light bulbs for mean lifetime. The first lot mean = 792 hours, S.D. = 35 hours, n = 25. The second lot mean = 776 hours, S.D. = 24 hours, n = 20. Determine with 95% confidence whether light bulbs from the first lot last longer than those from the second lot. Provide a statistic value.
(A) First lot lasts longer, z0 = -1.96
(B) No difference, z0 = 1.81
(C) No difference, t0 = 1.74
(D) First lot lasts longer, t0 = 1.96
Hypothesis testing in IE section of handbook page 198
ta values handbook page 21
Answer is (C)
Test H0: m1 = m2 vs. H1: m1 > m2
H0: m1 - m2 = 0 vs. H1: m1 - m2 > 0
Use formula for population standard deviation or variance unknown
Look up ta, n
a = 1 – 0.95 = 0.05
n = – 2 = 43 d.o.f.
t0.05, 43 = 1.96 from page 21 (n > 29)
Accept null hypothesis since statistic t0 < t0.05, 43
Bulbs from first lot do not last longer