Post on 04-Jun-2018
1
Fundamentals of Engineering Exam Review Series
Mathematics
Prof. Meredith Metzger Department of Mechanical Engineering
University of Utah
2
Overview 2
• 110 multiple choice questions total • 5 hrs 20 min to answer questions • slightly less than 3 minutes per question
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Overview 3
• 110 multiple choice questions total • 5 hrs 20 min to answer questions • slightly less than 3 minutes per question
Discipline Number of math questions % of test
Mechanical 6-9 5.5% - 8%
Electrical & Computer 11-17 10% - 15.5%
Civil 7-11 6% - 10%
Chemical 8-12 7% - 11%
Other 12-18 11% - 16%
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Mathematics Content 4
Discipline
Algebra &
Trigon
ometry
Analy5
c Geo
metry
Calculus
Line
ar Algeb
ra
Vector Ana
lysis
Diffe
ren5
al
Equa
5ons
Num
erical
Metho
ds
Complex Num
bers
Discrete
Mathe
ma5
cs
Roots o
f Equ
a5on
s
Mechanical ✔ ✔ ✔ ✔ ✔ ✔
Electrical & Computer ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔
Civil ✔ ✔ ✔ ✔
Chemical ✔ ✔ ✔ ✔
Other ✔ ✔ ✔ ✔ ✔ ✔
5
Mathematics Content 5
Discipline
Algebra &
Trigon
ometry
Analy5
c Geo
metry
Calculus
Line
ar Algeb
ra
Vector Ana
lysis
Diffe
ren5
al
Equa
5ons
Num
erical
Metho
ds
Complex Num
bers
Discrete
Mathe
ma5
cs
Roots o
f Equ
a5on
s
Mechanical ✔ ✔ ✔ ✔ ✔ ✔
Electrical & Computer ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔
Civil ✔ ✔ ✔ ✔
Chemical ✔ ✔ ✔ ✔
Other ✔ ✔ ✔ ✔ ✔ ✔
6
Permitted Calculators 6
• Casio FX-115 models
• HP 33 models
• HP 35 models
• TI-30x models
• TI-36x models
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Outline 7
I. Analytic Geometry
II. Algebra
III. Trigonometry
IV. Calculus
V. Differential Equations
VI. Linear Algebra and Vectors
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Analytic Geometry 8
• Equations and Curves
• Perimeter, Area, and Volume
• Conic Sections - Parabola - Hyperbola - Ellipse - Circle
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Straight Line (pg. 18) 9
10
Straight Line 10
11
Straight Line 11
12
Straight Line 12
13
Straight Line 13
14
Straight Line 14
15
Straight Line 15
16
Tangent Line to Circle 16
17
Tangent Line to Circle 17
18
Conic Sections (pgs. 22-23) 18
Writing equations for various conic sections
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Conic Sections 19
20
Conic Sections (pgs. 22-23) 20
21
Conic Sections 21
22
Conic Sections 22
23
Conic Sections 23
24
Conic Sections 24
25
Conic Sections 25
26
Conic Sections 26
27
Conic Sections 27
28
Conic Sections 28
29 Quadratic Surface (pg. 18) & Tangent Line to Circle (pg. 23)
29
30
Tangent Line to Circle 30
31
Tangent Line to Circle 31
32
Area (pgs. 20-21) 32
need to know: circle, rectangle, triangle
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Area 33
34
Area 34
35
Area 35
36
Area 36
37
Volume (pgs. 21-22) 37
38
Volume 38
39
Volume 39
40
Volume 40
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Volume 41
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Algebra 42
• Logarithms
• Complex Numbers
• Polar Coordinates
• Roots
• Progressions and Series - Arithmetic Progression - Geometric Progression - Properties of Series - Power Series
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Logarithms (pg. 19) 43
44
Logarithms 44
45
Logarithms 45
46
Logarithms 46
47
Logarithms 47
48
Logarithms 48
49
Logarithms 49
50
Complex Numbers (pg. 19) 50
51
Complex Numbers 51
52
Complex Numbers 52
53
Polar Coordinates (pg. 19) 53
54
Polar Coordinates 54
55
Polar Coordinates 55
56
Polar Coordinates 56
57
Polar Coordinates 57
58
Quadratic Equation (pg. 18) 58
59
Roots: Quadratic Equation 59
60
Roots: Quadratic Equation 60
61
Progressions and Series (pg. 26) 61
62
Progressions and Series 62
63
Progressions and Series 63
64
Progressions and Series 64
65
Progressions and Series 65
66
Trigonometry 66
• Degrees and Radians
• Plane Angles
• Triangles - Law of Sines - Law of Cosines
• Right Triangles
• General Triangles
• Trigonometric Identities
67
Angles – Basic Knowledge 67
radians = degrees * π/180
68
Triangles (pg. 19) 68
69
Triangles – Basic Knowledge 69
similar triangles
sides are proportional: b/e = c/f = a/d
70
Triangles 70
71
Triangles 71
72
Triangles 72
73
Triangles 73
74
Triangles 74
75
Identities (pg. 20) 75
76
Identities 76
77
Identities 77
78
Identities 78
79
Identities 79
80
Identities 80
81
Identities 81
82
Identities 82
83
Identities 83
84
Calculus 84
• Differential Calculus
• Critical Points
• Partial Derivatives
• Curvature
• Limits
• Integral Calculus
• Centroids and Moments of Inertia
• Taylor Series
85
Differential Calculus (pg. 23) 85
86
Derivative and Integral Table (pg. 25) 86
- Derivatives of polynomials missing - Product rule of differentiation - Integration by parts
87
Differential Calculus 87
88
Differential Calculus 88
89
Differential Calculus 89
90
Differential Calculus 90
91
Critical Points (pg. 23) 91
92
Critical Points 92
93
Critical Points 93
94
Critical Points 94
95
Critical Points 95
96
Partial Derivatives (pg. 23) 96
97
Partial Derivatives 97
98
Partial Derivatives 98
99
Curvature (pg. 24) 99
100
Curvature 100
101
Curvature 101
102
Limits (pg. 24) 102
103
Limits 103
104
Limits 104
105
Integral Calculus (pg. 24) 105
106
Integral Calculus 106
107
Integral Calculus 107
108
Derivative and Integral Table (pg. 25) 108
109
Centroids and Moments of Inertia (pg. 26) 109
110
Centroids and Moments of Inertia 110
111
Centroids and Moments of Inertia 111
112
Centroids and Moments of Inertia 112
113
Centroids and Moments of Inertia 113
114
Taylor Series (pg. 26) 114
115
Taylor Series 115
116
Taylor Series 116
117
Differential Equations 117
• Ordinary Linear Differential Equations
• 1st Order Homogenous ODEs
• 2nd Order Homogenous ODEs
• 1st Order Nonhomogeneous ODEs
• Fourier Transform
• Fourier Series
• Laplace Transform
118
Ordinary Linear Differential Eqn (pg. 27) 118
119
Ordinary Linear Differential Eqn 119
120
Ordinary Linear Differential Eqn 120
121
Ordinary Linear Differential Eqn 121
122
1st Order Homogeneous ODE (pg. 27) 122
123
1st Order Homogeneous ODE 123
124
1st Order Homogeneous ODE 124
125
2nd Order Homogeneous ODE (pg. 27) 125
126
2nd Order Homogeneous ODE 126
127
2nd Order Homogeneous ODE 127
128
1st Order Nonhomogeneous ODE (pg. 27) 128
129
1st Order Nonhomogeneous ODE 129
130
1st Order Nonhomogeneous ODE 130
131
Fourier Series (pg. 28) 131
132
Fourier Series 132
133
Fourier Series 133
134
Fourier Transform (pg. 27, 29) 134
135
Fourier Transform 135
136
Fourier Transform 136
137
Laplace Transform (pg. 30) 137
138
Laplace Transform 138
139
Laplace Transform 139
140
Laplace Transform 140
141
Laplace Transform 141
142
Linear Algebra & Vectors 142
• Matrix Arithmetic
• Matrix Transpose and Inverse
• Determinant of a Matrix
• Solving Systems of Linear Equations
• Vector Addition and Subtraction
• Vector Dot and Cross Products
• Vector Identities
• Gradient, Divergence, and Curl
143
Matrix Arithmetic (pg. 30) 143
144
Matrix Arithmetic 144
145
Matrix Arithmetic 145
146
Matrix Transpose and Inverse (pg. 30) 146
147
Matrix Transpose and Inverse 147
148
Matrix Transpose and Inverse 148
149
Matrix Transpose and Inverse 149
150
Matrix Transpose and Inverse 150
151
Matrix Transpose and Inverse 151
152
Determinants (pg. 31) 152
153
Determinants 153
154
Determinants 154
155
Determinants 155
156
Determinants 156
157
Systems of Linear Equations 157
158
Systems of Linear Equations 158
159
Vector Addition and Subtraction (pg. 31) 159
160
Vector Addition and Subtraction 160
161
Vector Addition and Subtraction 161
162
Vector Addition and Subtraction 162
163
Vector Addition and Subtraction 163
164
Vector Dot and Cross Products (pg. 31) 164
165
Vector Dot and Cross Products 165
166
Vector Dot and Cross Products 166
167
Vector Dot and Cross Products 167
168
Vector Dot and Cross Products 168
169
Vector Identities (pg. 31) 169
170
Vector Identities 170
171
Vector Identities 171
172
Gradient, Divergence, and Curl (pg. 31) 172
173
Gradient, Divergence, and Curl 173
174
Gradient, Divergence, and Curl 174
175
Gradient, Divergence, and Curl 175
176
Gradient, Divergence, and Curl 176
177
Gradient, Divergence, and Curl 177
178
Gradient, Divergence, and Curl 178