FE Mathematics ReviewDr. Scott MolitorAssociate ProfessorUndergraduate Program DirectorDepartment of Bioengineering

Topics coveredAnalytic geometry Equations of lines and curvesDistance, area and volumeTrigonometric identitiesIntegral calculusIntegrals and applicationsNumerical methodsDifferential equationsSolution and applicationsLaplace transformsDifference equations and Z transformsNumerical methodsDifferential calculusDerivatives and applicationsLimits and LHopitals ruleAlgebraComplex numbersMatrix arithmetic and determinantsVector arithmetic and applicationsProgressions and seriesNumerical methods for finding solutions of nonlinear equationsProbability and statisticsRules of probabilityCombinations and permutationsStatistical measures (mean, S.D., etc.)Probability density and distribution functionsConfidence intervalsHypothesis testingLinear regression

Tips for taking examUse the reference handbookKnow what it containsKnow what types of problems you can use it forKnow how to use it to solve problemsRefer to it frequentlyWork backwards when possibleFE exam is multiple choice with single correct answerPlug answers into problem when it is convenient to do soTry to work backwards to confirm your solution as often as possibleProgress from easiest to hardest problemSame number of points per problemCalculator tipsCheck the NCEES website to confirm your model is allowedAvoid using it to save time!Many answers do not require a calculator (fractions vs. decimals)

Equations of linesWhat 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

Equations of linesWhat 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 = 0Try using standard formHandbook pg 3: y = mx + bGiven (x1, y1) = (4, 0)Given (x2, y2) = (0, -6)Answer is (C)

Equations of linesWhat 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 = 0Work backwardsSubstitute (x1, y1) = (4, 0)Substitute (x2, y2) = (0, -6)See what worksAnswer is (C)

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

TrigonometryFor some angle q, csc q = -8/5. What is cos 2q?(A) 7/32(B) 1/4(C) 3/8(D) 5/8Use trigonometric identities on handbook page 5Answer is (A)Confirm with calculatorFirst find q = csc-1(-8/5)Then find cos 2q

Polar coordinatesWhat 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 coordinatesWhat 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 = 0Polar coordinate identities on handbook page 5Answer is (D)

Matrix identitiesFor 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 identitiesFor 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 7Answer is (C)Should know (A) and (D) are true from linear algebraAnswer (B) appears as matrix identity in handbook page 7Therefore can eliminate (C) as being true

Vector calculationsFor 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 calculationsFor 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) 216Vector products on handbook page 6Answer is (A)

Vector calculationsFor 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) 216Vector products on handbook page 6Answer 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!

Geometric progressionThe 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 progressionThe 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) 3/2Geometric progression on handbook page 7Answer is (B)

Geometric progressionThe 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) 3/2Geometric progression on handbook page 7Answer is (B)Confirm answer by calculating l2 and l6 with a = 1/5 and r = 3/2.

Roots of nonlinear equationsNewtons 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

Roots of nonlinear equationsNewtons 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) 4.0Newtons method on handbook page 13Answer is (D)

LimitsWhat is the limit of (1 e3x) / 4x as x 0?(A) -(B) -3/4(C) 0(D) 1/4

LimitsWhat is the limit of (1 e3x) / 4x as x 0?(A) -(B) -3/4(C) 0(D) 1/4LHopitals rule on handbook page 8Answer is (B)

LimitsWhat is the limit of (1 e3x) / 4x as x 0?(A) -(B) -3/4(C) 0(D) 1/4LHopitals rule on handbook page 8Answer is (B)You should apply LHopitals 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 0.001.

Application of derivativesThe 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) 0.034 m3 / min(B) 0.52 m3 / min(C) 0.63 m3 / min(D) 0.84 m3 / min

Application of derivativesThe 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) 0.034 m3 / min(B) 0.52 m3 / min(C) 0.63 m3 / min(D) 0.84 m3 / minDerivatives on handbook page 9; volume of sphere on handbook page 10Answer is (C)

Application of derivativesThe 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) 0.034 m3 / min(B) 0.52 m3 / min(C) 0.63 m3 / min(D) 0.84 m3 / minDerivatives on handbook page 9; volume of sphere on handbook page 10Answer is (C)Convert cm to m, convert diameter to radius, and confirm final units are correct.

Evaluating integralsEvaluate 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

Evaluating integralsEvaluate 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 + CApply integration by parts on handbook page 9Answer is (B)

Evaluating integralsEvaluate 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 + CAlternative method is to differentiate answersAnswer is (B)

Applications of integralsWhat 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

Applications of integralsWhat 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) 4Need absolute value because sin x is negative over interval [p, 2p]Answer is (C)

Differential equationsWhat 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

Differential equationsWhat 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 + C2e4xSolving 2nd order differential eqns on handbook page 12Answer is (B)

Differential equationsWhat 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 + C2e4xSolving 2nd order differential eqns on handbook page 12Answer 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).

Laplace transformsFind 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 transformsFind 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)

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

Probability of an outcomeA marksman can hit a bulls-eye 3 out of 4 shots. What is the probability he will hit a bulls-eye with at least 1 of his next 3 shots?(A) 3/4(B) 15/16(C) 31/32(D) 63/64Answer is (D)Let H = hit, M = miss, Prob(H) = , Prob(M) = Use combinations for next three shotsFind 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 = 1/64Answer is 1 1/64 = 63/64

Normal distributionExam 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) 0.4196(B) 0.4837(C) 0.5161(D) 0.6455

Normal distributionExam 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) 0.4196(B) 0.4837(C) 0.5161(D) 0.6455Standard normal tables on handbook page 20Answer 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 = -0.73 -0.7(80 72) / 11 = 0.64 0.6Prob(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 tableProb(65 < X < 80) 0.7257 0.2420 = 0.4837

Confidence intervalsWhat 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 intervalsWhat 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 0.86Confidence intervals on handbook page 19ta values handbook page 21Answer is (A)Use formula for population standard deviation unknownFormula 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 = 2.064 on page 21Calculate confidence interval 73 (2.064) (11) / 25Answer is 73 4.54

Hypothesis testingYou 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

Hypothesis testingYou 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 = 1.96Hypothesis testing in IE section of handbook page 198ta values handbook page 21Answer is (C)Test H0: m1 = m2 vs. H1: m1 > m2H0: m1 - m2 = 0 vs. H1: m1 - m2 > 0Use formula for population standard deviation or variance unknown

Look up ta, n a = 1 0.95 = 0.05 n = 25 + 20 2 = 43 d.o.f. t0.05, 43 = 1.96 from page 21 (n > 29)Accept null hypothesis since statistic t0 < t0.05, 43Bulbs from first lot do not last longer