+ fe mathematics review dr. omar meza assistant professor department of mechanical engineering
TRANSCRIPT
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+FE FE Mathematics Mathematics ReviewReview
Dr. Omar MezaDr. Omar MezaAssistant ProfessorAssistant ProfessorDepartment of Mechanical EngineeringDepartment of Mechanical Engineering
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+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
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+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)
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+Equations of linesEquations of lines
Handbook page:Handbook page:
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+Equations of linesEquations of lines
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+Equations of linesEquations of lines
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+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
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+Equations of linesEquations of lines
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)
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+Equations of linesEquations of lines
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
Answer is (C)Answer is (C)
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+Equations of linesEquations of lines
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+Equations of linesEquations of lines
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+Equations of linesEquations of lines
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+Quadratic Equation
Handbook page:
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+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)
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+Quadratic EquationQuadratic Equation
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+Equations of curvesEquations of curves
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+Equations of curvesEquations of curves
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+Equations of curvesEquations of curves
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+Equations of curvesEquations of curves
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+Equations of curvesEquations of curves
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+Equations of curvesEquations of curves
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+Equations of curvesEquations of curves
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+Equations of curvesEquations of curves
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+Equations of curvesEquations of curves
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+Equations of curvesEquations of curves
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+Equations of curvesEquations of curves
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+Equations of curvesEquations of curves
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+Equations of curvesEquations of curves
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+Equations of curvesEquations of curves
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+Equations of curvesEquations of curves
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+Equations of curvesEquations of curves
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+LogarithmsLogarithms
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+
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)
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+LogarithmsLogarithms
88.1=)8ln()50ln(
=50ln8
Answer is (D)Answer is (D)
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+TrigonometryTrigonometry
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+Trigonometry
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+TrigonometryTrigonometry
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+TrigonometryTrigonometry
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+TrigonometryTrigonometry
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)
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+TrigonometryTrigonometry
θ2
θθ
θ
θθ
θ
2
22
2
22
2
cos
cos+cos
)cos1
(
1+))(sin
sincos(
Answer is (C)Answer is (C)
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+TrigonometryTrigonometry
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+Complex NumbersComplex Numbers
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+Complex NumbersComplex Numbers
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+Complex NumbersComplex Numbers
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+Complex NumbersComplex Numbers
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+Polar coordinatesPolar coordinates
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+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
Answer is (D)Answer is (D)
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+Polar coordinatesPolar coordinates
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+MatricesMatrices
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+MatricesMatrices
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+MatricesMatrices
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+MatricesMatrices
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+MatricesMatrices
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+MatricesMatrices
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+MatricesMatrices
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+MatricesMatrices
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+MatricesMatrices
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+MatricesMatrices
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+VectorVector
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+VectorVector
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+VectorVector
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+VectorVector
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+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
Answer is (A)Answer is (A)
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+Vector calculationsVector calculations
Answer is (D)Answer is (D)
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+Vector calculationsVector calculations
Answer is (A)Answer is (A)
(-16- 8)i – (-8+16)j + (2+8)k
-24i -8j + 10k
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+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.
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+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
Answer is (D)Answer is (D)
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+Application of derivativesApplication of derivatives
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+Application of derivativesApplication of derivatives
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+Application of derivativesApplication of derivatives
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+Application of derivativesApplication of derivatives
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+Application of derivativesApplication of derivatives
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+Application of derivativesApplication of derivatives
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+Application of derivativesApplication of derivatives
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+Application of derivativesApplication of derivatives
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+Application of derivativesApplication of derivatives
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+Application of derivativesApplication of derivatives
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+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.
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+Application of derivativesApplication of derivatives
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.
Answer is (C)Answer is (C)
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+Evaluating integralsEvaluating integrals
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+Evaluating integralsEvaluating integrals
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+Evaluating integralsEvaluating integrals
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+Evaluating integralsEvaluating integrals
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+Evaluating integralsEvaluating integrals
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+Evaluating integralsEvaluating integrals
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+Evaluating integralsEvaluating integrals
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+Evaluating integralsEvaluating integrals
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+Evaluating integralsEvaluating integrals
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+Evaluating integralsEvaluating integrals
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+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
Answer is (B)Answer is (B)
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+Evaluating integralsEvaluating integrals
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
Answer is (B)Answer is (B)
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+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/
Answer is (C)Answer is (C)
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+Differential EquationsDifferential Equations
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+Differential EquationsDifferential Equations
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+Differential EquationsDifferential Equations
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+Differential EquationsDifferential Equations
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+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
Answer is (B)Answer is (B)
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+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
Answer is (A)Answer is (A)
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+
Preguntas? Preguntas? Comentarios?Comentarios?
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+
Muchas Muchas Gracias !Gracias !