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
