+ FE Mathematics Review Dr. Omar Meza Assistant Professor Department of Mechanical Engineering

+FE FE Mathematics Mathematics ReviewReview

Dr. Omar MezaDr. Omar MezaAssistant ProfessorAssistant ProfessorDepartment of Mechanical EngineeringDepartment of Mechanical Engineering

+Topics coveredTopics covered

Analytic geometry Analytic geometry Equations of lines and curvesEquations of lines and curves Distance, area and volumeDistance, area and volume Trigonometric identitiesTrigonometric identities

AlgebraAlgebra Complex numbersComplex numbers Matrix arithmetic and Matrix arithmetic and

determinantsdeterminants Vector arithmetic and Vector arithmetic and

applicationsapplications Progressions and seriesProgressions and series Numerical methods for Numerical methods for

finding solutions of nonlinear finding solutions of nonlinear equationsequations

Differential calculusDifferential calculus Derivatives and applicationsDerivatives and applications Limits and L’Hopital’s ruleLimits and L’Hopital’s rule

Integral calculusIntegral calculus Integrals and applicationsIntegrals and applications Numerical methodsNumerical methods

Differential equationsDifferential equations Solution and applicationsSolution and applications Laplace transformsLaplace transforms

+Tips for taking examTips for taking exam

Use the reference handbookUse the reference handbook Know what it containsKnow what it contains Know what types of problems you can use it forKnow what types of problems you can use it for Know how to use it to solve problemsKnow how to use it to solve problems Refer to it frequentlyRefer to it frequently

Work backwards when possibleWork backwards when possible FE exam is multiple choice with single correct answerFE exam is multiple choice with single correct answer Plug answers into problem when it is convenient to do soPlug answers into problem when it is convenient to do so Try to work backwards to confirm your solution as often as Try to work backwards to confirm your solution as often as

possiblepossible Progress from easiest to hardest problemProgress from easiest to hardest problem

Same number of points per problemSame number of points per problem Calculator tipsCalculator tips

Check the NCEES website to confirm your model is allowedCheck the NCEES website to confirm your model is allowed Avoid using it to save time!Avoid using it to save time! Many answers do not require a calculator (fractions vs. decimals)Many answers do not require a calculator (fractions vs. decimals)

+Equations of linesEquations of lines

Handbook page:Handbook page:



+Equations of lines Equations of lines

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

-0--1--2--3--4--5--6-

1 2 3 4 5


12y2x3=0

12x3=y2

6x23

=y

6=b

23

=400

=xxyy

=m

b+xm=y

12

12

-.-.

-..

-.

-

-

--6

-

-

.

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

Try using standard formTry using standard form Handbook pg 3: y = mx + bHandbook pg 3: y = mx + b Given (x1, y1) = (4, 0)Given (x1, y1) = (4, 0) Given (x2, y2) = (0, -6)Given (x2, y2) = (0, -6)

Answer is (C)Answer is (C)


012)6(203)C(

024120243)D(

0120243)C(

026180342)B(

010180342)A(

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

Work backwardsWork backwards Substitute (x1, y1) = (4, 0)Substitute (x1, y1) = (4, 0) Substitute (x2, y2) = (0, -6)Substitute (x2, y2) = (0, -6) See what worksSee what works

Alternative SolutionAlternative Solution





+Quadratic Equation

Handbook page:

+Quadratic Equation

Handbook page:

3-5x+2x=f(x)ofrootstheareWhat 2

A) 1, 2; B) 3, 2; C) 0.5,-3; D) -0.5, -3

Answer is (C)

+Quadratic EquationQuadratic Equation

+Equations of curvesEquations of curves
















+LogarithmsLogarithms

+

xy2)D(

xy8686.0)C(

xy5.0)B(

xy/2)A(

LogarithmsLogarithms

xy2

)000006.2(xy

)3891.7ln(xy

)3891.7ln( xy

≈

Answer is (D)Answer is (D)

+LogarithmsLogarithms

88.1=)8ln()50ln(

=50ln8


+TrigonometryTrigonometry

+Trigonometry




32

7

32

2512cos

64

2521

8

5212cos

csc

1212cos

sin212cos

sin

1csc

2

2

2

2

For some angle For some angle , csc , csc = -8/5. = -8/5. What is cos 2 What is cos 2??

Use trigonometric identities Use trigonometric identities on handbook.on handbook.

Confirm with calculatorConfirm with calculator First find First find = csc = csc-1-1(-8/5)(-8/5) Then find cos 2Then find cos 2

(A) 7/32(A) 7/32(B) 1/4(B) 1/4(C) 3/8(C) 3/8(D) 5/8(D) 5/8

Answer is (A)Answer is (A)


θ2

θθ

θ

θθ

θ

2

22

2

22

2

cos

cos+cos

)cos1

(

1+))(sin

sincos(



+Complex NumbersComplex Numbers




+Polar coordinatesPolar coordinates


0=y+yx+xx

yx=yx+xxy

1=y+x

))xy((tantan1=)y+x(

tan1=rxy

=tan),xy(tan=

y+x=r

22224

22224

2

222

12222

22

1

22

-

-

-

-

θ-

θθ

What is rectangular form of What is rectangular form of the polar equation rthe polar equation r22 = 1 – = 1 – tantan22 ?? (A) –x(A) –x22 + x + x44yy22 + y + y22 = 0 = 0 (B) x(B) x22 + x + x22yy22 - y - y22 - y - y44 = 0 = 0 (C) –x(C) –x44 + y + y22 = 0 = 0 (D) x(D) x44 – x – x22 + x + x22yy22 + y + y22 = 0 = 0

Polar coordinate identities Polar coordinate identities on handbookon handbook



+MatricesMatrices

+MatricesMatrices

+MatricesMatrices

+MatricesMatrices

+MatricesMatrices

+MatricesMatrices

+MatricesMatrices

+MatricesMatrices

+MatricesMatrices

+MatricesMatrices

+VectorVector

+VectorVector

+VectorVector

+VectorVector

+Vector calculationsVector calculations

0)2(1048)2(6)CB(A

)k2j4i2()k10j8i6()CB(A

k2j4i2CB

)3241(k)3351(j)4352(iCB

543

321

kji

CB

For three vectorsFor three vectorsA = 6i + 8j + 10kA = 6i + 8j + 10kB = i + 2j + 3kB = i + 2j + 3kC = 3i + 4j + 5k, what is the C = 3i + 4j + 5k, what is the product A·(B x C)?product A·(B x C)? (A) 0(A) 0 (B) 64(B) 64 (C) 80(C) 80 (D) 216(D) 216

Vector products on Vector products on handbookhandbook






(-16- 8)i – (-8+16)j + (2+8)k

-24i -8j + 10k

+Geometric ProgressionGeometric Progression

5

1al

10

3

2

3al

2

3

16

81r

16

81

10/3

160/243r

ar

ar

l

l

160

243l,

10

3l

arl

1

2

4

45

2

6

62

1nn

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

Geometric progression on Geometric progression on handbookhandbook

Answer is (B)Answer is (B) Confirm answer by calculating lConfirm answer by calculating l22

and land l66 with a = 1/5 and r = 3/2. with a = 1/5 and r = 3/2.

+Roots of nonlinear equationsRoots of nonlinear equations

0.446.7

91.1273.5x

)273.5(2

1)273.5(73.5x

73.566.14

73.5233.9x

)233.9(2

1)233.9(33.9x

33.9x

)2x(2)x(f

1)2x()x(f

)x(f

)x(fxx

3

2

3

2

2

2

1

2

n

nn1n

Newton’s method is being Newton’s method is being used to find the roots of the used to find the roots of the equation f(x) = (x – 2)equation f(x) = (x – 2)22 – 1. – 1. Find the 3Find the 3rdrd approximation if approximation if the 1the 1stst approximation of the approximation of the root is 9.33root is 9.33 (A) 1.0(A) 1.0 (B) 2.0(B) 2.0 (C) 3.0(C) 3.0 (D) 4.0(D) 4.0

Newton’s method on Newton’s method on handbookhandbook


+Application of derivativesApplication of derivatives










+LimitsLimits

4

3

4

13

4

e3lim

4

e3lim

x4

e1lim

)x('g

)x('flimtry,

0

0

)x(g

)x(flimif

?0

0

0

11

04

e1

x4

e1lim

x3

0x

x3

0x

x3

0x

0x0x

03x3

0x

What is the limit of (1 – eWhat is the limit of (1 – e3x3x) / ) / 4x as x 4x as x 0? 0? (A) -(A) -∞∞ (B) -3/4(B) -3/4 (C) 0(C) 0 (D) 1/4(D) 1/4

L’Hopital’s rule on L’Hopital’s rule on handbookhandbook

Answer is (B)Answer is (B)

You should apply L’Hopital’s rule You should apply L’Hopital’s rule iteratively until you find limit of iteratively until you find limit of f(x) / g(x) that does not equal 0 / 0.f(x) / g(x) that does not equal 0 / 0.

You can also use your calculator to You can also use your calculator to confirm the answer, substitute a confirm the answer, substitute a small value of x = 0.01 or 0.001.small value of x = 0.01 or 0.001.


min

m63.0

dt

dV

min

m2.0m5.04

dt

dVdt

drr4

dt

dVdt

dr

dr

dV

dt

dV

r3

4)r(V

3

2

2

3

The radius of a snowball The radius of a snowball

rolling down a hill is rolling down a hill is increasing at a rate of 20 increasing at a rate of 20 cm / min. How fast is its cm / min. How fast is its volume increasing when volume increasing when the diameter is 1 m?the diameter is 1 m? (A) 0.034 m(A) 0.034 m33 / min / min (B) 0.52 m(B) 0.52 m33 / min / min (C) 0.63 m(C) 0.63 m33 / min / min (D) 0.84 m(D) 0.84 m33 / min / min

Derivatives on handbook; Derivatives on handbook; volume of sphere on volume of sphere on handbook page 10handbook page 10 Convert cm to m, convert diameter Convert cm to m, convert diameter

to radius, and confirm final units to radius, and confirm final units are correct.are correct.


+Evaluating integralsEvaluating integrals











xcos3

1dxxsinxcos

xcosdxxsinxcos3

dxxsinxcos2xcosdxxsinxcos

duvvudvu

xcosv

dxxsindv

dxxsinxcos2du

xcosu

32

32

232

2

Evaluate the indefinite Evaluate the indefinite integral of f(x) = cosintegral of f(x) = cos22x sin xx sin x (A) -2/3 sin(A) -2/3 sin33x + Cx + C (B) -1/3 cos(B) -1/3 cos33x + Cx + C (C) 1/3 sin(C) 1/3 sin33x + Cx + C (D) 1/2 sin(D) 1/2 sin22x cosx cos22x + Cx + C

Apply integration by parts Apply integration by parts on handbookon handbook



xcosxsinxcosxsin)Cxcosxsin2

1(

dx

d)D(

xcosxsin)Cxsin3

1(

dx

d)C(

xsinxcos)Cxcos3

1(

dx

d)B(

xcosxsin2)Cxsin3

2(

dx

d)A(

3322

23

23

23

Evaluate the indefinite Evaluate the indefinite integral of f(x) = cosintegral of f(x) = cos22x sin xx sin x (A) -2/3 sin(A) -2/3 sin33x + Cx + C (B) -1/3 cos(B) -1/3 cos33x + Cx + C (C) 1/3 sin(C) 1/3 sin33x + Cx + C (D) 1/2 sin(D) 1/2 sin22x cosx cos22x + Cx + C

Alternative method is to Alternative method is to differentiate answersdifferentiate answers


+Applications of integralsApplications of integrals

/2

2

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

Need absolute value Need absolute value because sin x is negative because sin x is negative over interval [over interval [, 2, 2]]

3)1(10)1(area

xcosxcosarea

dxxsindxxsinarea

dxxsinarea

22/

2

2/

2

2/


+Differential EquationsDifferential Equations




+Differential equationsDifferential equations

x421

2

2

e)xCC(y

41644r

0r16r42r

16b,4a

0y16y42y

0y16y8y

What is the general solution What is the general solution

to the differential equationto the differential equationy’’ – 8y’ + 16y = 0?y’’ – 8y’ + 16y = 0? (A) y = C(A) y = C11ee4x4x

(B) y = (C(B) y = (C11 + C + C22x)ex)e4x4x

(C) y = C(C) y = C11ee-4x-4x + C + C11ee4x4x

(D) y = C(D) y = C11ee2x2x + C + C22ee4x4x

Solving 2nd order Solving 2nd order differential eqns on differential eqns on handbookhandbook


+Laplace transformsLaplace transforms

222

222

222

22t0

2

22

s1s

1)s(F

s)s(F)1s(

s)s(F)s(Fs

stsinetsin

)s(F)t(f

)s(Fs)t(f

)0(fs)0(fs)s(Fs)t(f

Find the Laplace transform of Find the Laplace transform of the equation f”(t) + f(t) = sin the equation f”(t) + f(t) = sin t t where f(0) and f’(0) = 0where f(0) and f’(0) = 0 (A) F(s) = (A) F(s) = / [(1 + s / [(1 + s22)(s)(s22 + + 22)])] (B) F(s) = (B) F(s) = / [(1 + s / [(1 + s22)(s)(s22 - - 22)])] (C) F(s) = (C) F(s) = / [(1 - s / [(1 - s22)(s)(s22 + + 22)])] (D) F(s) = s / [(1 - s(D) F(s) = s / [(1 - s22)(s)(s22 + + 22)])]

Laplace transforms on Laplace transforms on handbookhandbook


+

Preguntas? Preguntas? Comentarios?Comentarios?

+

Muchas Muchas Gracias !Gracias !

Documents

+ FE Mathematics Review Dr. Omar Meza Assistant Professor Department of Mechanical Engineering