1
Fundamentals of Engineering Exam
Review Series
Mathematics
Prof. Meredith Metzger
Department of Mechanical Engineering
University of Utah
2
Overview
• 110 multiple choice questions total
• 5 hrs 20 min to answer questions
• slightly less than 3 minutes per question
3
Overview
• 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%
4
Algebra & Trigonometry Analy5c Geometry Calculus Linear Algebra Vector Analysis Differen5al Equa5ons Numerical Methods Complex Numbers Discrete Mathema5cs Roots of Equa5ons Mathematics Content
Mechanical ✔
✔
✔
✔
✔
✔
Electrical & Computer ✔
✔
✔
✔
✔
✔
✔
✔
Civil ✔
✔
✔
✔
Chemical ✔
✔
✔
✔
Other ✔
✔
✔
✔
✔
✔
Discipline 5
Linear Algebra Vector Analysis Differen5al Equa5ons Numerical Methods Complex Numbers Discrete Mathema5cs Roots of Equa5ons Mechanical ✔
✔
✔
✔
✔
✔
Electrical & Computer ✔
✔
✔
✔
✔
✔
✔
✔
Civil ✔
✔
✔
✔
Chemical ✔
✔
✔
✔
Other ✔
✔
✔
✔
✔
✔
Discipline Algebra & Trigonometry Analy5c Geometry Calculus Mathematics Content
6
Permitted Calculators
• Casio FX-115 models
• HP 33 models
• HP 35 models
• TI-30x models
• TI-36x models
7
Outline
I.
Analytic Geometry
II. Algebra
III. Trigonometry
IV. Calculus
V. Differential Equations
VI. Linear Algebra and Vectors
8
Analytic Geometry
• Equations and Curves
• Perimeter, Area, and Volume
• Conic Sections
- Parabola
- Hyperbola
- Ellipse
- Circle
9
Straight Line (pg. 18)
10
Straight Line
11
Straight Line
12
Straight Line
13
Straight Line
14
Straight Line
15
Straight Line
16
Tangent Line to Circle
17
Tangent Line to Circle
18
Conic Sections (pgs. 22-23)
Writing equations for various conic sections
19
Conic Sections
20
Conic Sections (pgs. 22-23)
21
Conic Sections
22
Conic Sections
23
Conic Sections
24
Conic Sections
25
Conic Sections
26
Conic Sections
27
Conic Sections
28
Conic Sections
29
Quadratic Surface (pg. 18) &
Tangent Line to Circle (pg. 23)
30
Tangent Line to Circle
31
Tangent Line to Circle
32
Area (pgs. 20-21)
need to know: circle, rectangle, triangle
33
Area
34
Area
35
Area
36
Area
37
Volume (pgs. 21-22)
38
Volume
39
Volume
40
Volume
41
Volume
42
Algebra
•
Logarithms
• Complex Numbers
• Polar Coordinates
• Roots
• Progressions and Series
- Arithmetic Progression
- Geometric Progression
- Properties of Series
- Power Series
43
Logarithms (pg. 19)
44
Logarithms
45
Logarithms
46
Logarithms
47
Logarithms
48
Logarithms
49
Logarithms
50
Complex Numbers (pg. 19)
51
Complex Numbers
52
Complex Numbers
53
Polar Coordinates (pg. 19)
54
Polar Coordinates
55
Polar Coordinates
56
Polar Coordinates
57
Polar Coordinates
58
Quadratic Equation (pg. 18)
59
Roots: Quadratic Equation
60
Roots: Quadratic Equation
61
Progressions and Series (pg. 26)
62
Progressions and Series
63
Progressions and Series
64
Progressions and Series
65
Progressions and Series
66
Trigonometry
• Degrees and Radians
• Plane Angles
• Triangles
- Law of Sines
- Law of Cosines
• Right Triangles
• General Triangles
• Trigonometric Identities
67
Angles – Basic Knowledge
radians = degrees * π/180
68
Triangles (pg. 19)
69
Triangles – Basic Knowledge
similar triangles
sides are proportional: b/e = c/f = a/d
70
Triangles
71
Triangles
72
Triangles
73
Triangles
74
Triangles
75
Identities (pg. 20)
76
Identities
77
Identities
78
Identities
79
Identities
80
Identities
81
Identities
82
Identities
83
Identities
84
Calculus
• Differential Calculus
• Critical Points
• Partial Derivatives
• Curvature
• Limits
• Integral Calculus
• Centroids and Moments of Inertia
• Taylor Series
85
Differential Calculus (pg. 23)
86
Derivative and Integral Table (pg. 25)
- Derivatives of polynomials missing
- Product rule of differentiation
- Integration by parts
87
Differential Calculus
88
Differential Calculus
89
Differential Calculus
90
Differential Calculus
91
Critical Points (pg. 23)
92
Critical Points
93
Critical Points
94
Critical Points
95
Critical Points
96
Partial Derivatives (pg. 23)
97
Partial Derivatives
98
Partial Derivatives
99
Curvature (pg. 24)
10
0
Curvature
10
1
Curvature
10
2
Limits (pg. 24)
10
3
Limits
10
4
Limits
10
5
Integral Calculus (pg. 24)
10
6
Integral Calculus
10
7
Integral Calculus
10
8
Derivative and Integral Table (pg. 25)
10
9
Centroids and Moments of Inertia (pg. 26)
11
0
Centroids and Moments of Inertia
11
1
Centroids and Moments of Inertia
11
2
Centroids and Moments of Inertia
11
3
Centroids and Moments of Inertia
11
4
Taylor Series (pg. 26)
11
5
Taylor Series
11
6
Taylor Series
11
7
Differential Equations
• Ordinary Linear Differential Equations
• 1st Order Homogenous ODEs
• 2nd Order Homogenous ODEs
• 1st Order Nonhomogeneous ODEs
• Fourier Transform
• Fourier Series
• Laplace Transform
11
8
Ordinary Linear Differential Eqn (pg. 27)
11
9
Ordinary Linear Differential Eqn
12
0
Ordinary Linear Differential Eqn
12
1
Ordinary Linear Differential Eqn
12
2
1st Order Homogeneous ODE (pg. 27)
12
3
1st Order Homogeneous ODE
12
4
1st Order Homogeneous ODE
12
5
2nd Order Homogeneous ODE (pg. 27)
12
6
2nd Order Homogeneous ODE
12
7
2nd Order Homogeneous ODE
12
8
1st Order Nonhomogeneous ODE (pg. 27)
12
9
1st Order Nonhomogeneous ODE
13
0
1st Order Nonhomogeneous ODE
13
1
Fourier Series (pg. 28)
13
2
Fourier Series
13
3
Fourier Series
13
4
Fourier Transform (pg. 27, 29)
13
5
Fourier Transform
13
6
Fourier Transform
13
7
Laplace Transform (pg. 30)
13
8
Laplace Transform
13
9
Laplace Transform
14
0
Laplace Transform
14
1
Laplace Transform
14
2
Linear Algebra & Vectors
• 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
14
3
Matrix Arithmetic (pg. 30)
14
4
Matrix Arithmetic
14
5
Matrix Arithmetic
14
6
Matrix Transpose and Inverse (pg. 30)
14
7
Matrix Transpose and Inverse
14
8
Matrix Transpose and Inverse
14
9
Matrix Transpose and Inverse
15
0
Matrix Transpose and Inverse
15
1
Matrix Transpose and Inverse
15
2
Determinants (pg. 31)
15
3
Determinants
15
4
Determinants
15
5
Determinants
15
6
Determinants
15
7
Systems of Linear Equations
15
8
Systems of Linear Equations
15
9
Vector Addition and Subtraction (pg. 31)
160
16
0
Vector Addition and Subtraction
161
16
1
Vector Addition and Subtraction
162
16
2
Vector Addition and Subtraction
163
16
3
Vector Addition and Subtraction
164
16
4
Vector Dot and Cross Products (pg. 31)
165
16
5
Vector Dot and Cross Products
166
16
6
Vector Dot and Cross Products
167
16
7
Vector Dot and Cross Products
168
16
8
Vector Dot and Cross Products
169
16
9
Vector Identities (pg. 31)
170
17
0
Vector Identities
171
17
1
Vector Identities
172
17
2
Gradient, Divergence, and Curl (pg. 31)
173
17
3
Gradient, Divergence, and Curl
174
17
4
Gradient, Divergence, and Curl
175
17
5
Gradient, Divergence, and Curl
176
17
6
Gradient, Divergence, and Curl
177
17
7
Gradient, Divergence, and Curl
178
17
8
Gradient, Divergence, and Curl
0
