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Mathematics 1 – Single Variable Calculus (3 credits), Differential Equations (1 credit) |
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Topic |
Calculus |
Difference Equations |
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Dynamical Systems (1 week) |
Functions Initial Value Problem |
Sequences Representations of a DDS Numerical Solutions Analytical Solutions |
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First-Order, Linear, Difference and Differential Equations1 (3 weeks) |
Antiderivatives Homogeneous Equations Non-Homogeneous Equations Slope Fields Autonomous Equations Phase Diagrams Equilibria and Stability |
Homogeneous Equations Non-Homogeneous Equations Equilibria and Stability Undetermined Coefficients Exponential Functions Polynomials Long Term Behavior of Solutions |
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Systems of Difference Equations2 (1 week) |
Numerical Solutions Graphs of Solutions Phase Diagrams Equilibria and Stability Long Term Behavior of Solutions |
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Transition to Calculus (1 week) |
Discrete and Continuous Functions Secant Slopes Riemann Sums The Fundamental Theorem |
Sequences as Functions Discrete Sampling of Continuous Functions Difference Tables Increasing and Decreasing Sequences Concavity Sums and Series Rates of Change, Change, and Accumulation of Change Estimating Areas, Volumes, and Length |
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Review of Functions (1 week) |
Linear Functions and Slope Inverse Functions Trig Functions Exponential and Log Functions |
Polynomial Functions Exponential Functions Logarithms |
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Limits and Continuity (2 weeks) |
Limits Informally Limits Formally Limits Involving Infinity Continuity |
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Derivatives (5 weeks) |
Tangent Lines Rates of Change The Derivative of a Function The Chain Rule Implicit Differentiation Related Rates The Shape of a Graph Extreme Values |
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1 ILAP on Yellowstone Bison or Planning for Your Retirement 2 ILAP on Deer and Cougar or Lake Pollution |
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Mathematics 2 – Single Variable Calculus (3 credits), Linear Algebra (1 credit) |
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Topic |
Calculus |
Linear Algebra |
Difference and Differential Equations |
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Dynamical Systems (1 week) |
Separation of Variables Euler’s Method |
Separation of Variables Euler’s Method |
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Applications of the Derivative (3 weeks) |
Optimization The Mean Value Theorem Linearization and Differentials Newton’s Method |
Corollaries to the Mean Value Theorem Graphical Solutions of Autonomous Diff Eqs Phase Lines Equilibria and Stability Long-Term Behavior |
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Integration (3 weeks) |
Indefinite Integrals u-Substitutions Riemann Sums Definite Integrals Fundamental Theorem Numerical Integration |
Antiderivatives Estimating with Finite Sums |
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Applications of Integrals3 (2 weeks) |
Areas, Volumes, and Arc Lengths Forces and Work Moments and Centers |
Differential Forms for Applications of Integrals |
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Transcendental Functions (1 week) |
Logarithms Exponential Fns. Inverse Trig Fns. |
Separable Diff Eqs Initial Value Problems |
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Systems of Linear Equations (2 weeks) |
Vectors and Matrices Gauss Elimination Singular and Non-Singular Matrices Determinants and Inverses Existence and Uniqueness of Solutions |
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Eigenvalues and Eigenvectors4 (2 weeks) |
Finding Eigenvalues with Determinants Finding Eigenvectors |
Solving Systems of Linear Difference Eqns Equilibria and Stability Long-Term Behavior of Solutions |
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3 Project on Bungee Cord or Compound Bow 4 ILAP on Lake Pollution (Part 1) |
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Mathematics 3 – Multivariable Calculus (4 credits) & Linear Algebra (1 credit) |
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Topic |
Calculus |
Differential Equations |
Linear Algebra |
Probability and Statistics |
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Overview of Calculus Topics (1 week) |
Review MVTs, FTIC Important Limits Improper Integrals Conics Sections Polar Coordinates |
Monte Carlo Method for Numerical Integration |
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Vectors in Two Dimensions (1.5 weeks) |
Dot Product Parametric Equations Equations of Motion Polar Representation |
Vectors Dot Product Applied to Linear Regression |
Dot Product Applied to Linear Regression and Correlation |
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Vectors in Three Dimensions Mappings5 (1.5 weeks) |
Cross Product Equations of Lines and Planes Equations of Motion Projections from 3 to 2 Dimensions |
Vectors Mappings Linear Transformations |
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Surfaces and Partial Derivatives6 (4 weeks) |
Surfaces Cylindrical, Spherical Coordinate Systems Partial Derivatives Tangent Planes and Linearizations Directional Derivative Optimization Lagrange Multipliers |
Introduction to PDE |
Lagrange Multiplier Used to Introduce Linear Programming Hessian of a Function |
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Multiple Integrals7 (2 weeks) |
Multiple Integration Setups and Applications Transformations to Alternate Coordinate Systems |
Jacobian of a Transformation |
Monte Carlo Method and Error Analysis Computation of Probabilities Moments As Means and Standard Deviations |
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Line and Surface Integrals (2.5 weeks) |
Work and Flux Green’s Theorem Stokes’ and Div. Theorem (briefly) |
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Series (1.5 weeks) |
Taylor Series Euler’s Formula Fourier Series |
Introduction to the Heat Equation |
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5 Project on CAT Scan Technology 6 Project on Contour Maps 7 Project on Error Analysis |
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Mathematics 4 – Differential Equations (2 credits), Linear Algebra (1 credit), and Probability and Statistics (2 credits) |
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Topic |
Calculus |
Differential Equations |
Linear Algebra |
Probability and Statistics |
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2nd Order Linear DE (1 week) |
Application of Taylor Series |
2nd Order Linear DEs Intro to Systems of Linear DEs |
Linear Independence of Solutions |
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Heat Equation (1 weeks) |
Application of Fourier Series |
Solution of PDE Heat Equation (BVP) |
Eigenfunctions |
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Systems of Difference & Differential Equations8 (3 weeks) |
Application of Geome-tric Series |
Equilibria and Stability Linear Systems Nonlinear Systems Phase Diagrams |
Eigenvalues and Eigenvectors Equilibria and Stability Jacobian Vector Space |
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Systems of Differential Equations (2 weeks) |
Linear Systems Nonlinear Systems Phase Diagrams Equilibria and Stability |
Eigenvalues and Eigenvectors Linearization and Jacobian Vector Space Basis |
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Vector Spaces and Linear Transforma- Tions8 (2 weeks) |
Relation between null space and solution set for homogeneous linear differential equation |
Kernel of a Transformation Eigenvalues and Eigenvectors Spanning Sets Diagonalization |
Markov Chains Leslie Matrices |
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Statistics9 (1.5 weeks) |
Curve Fitting Using Least Squares |
Measures of Location and Dispersion Exploratory Data Analysis Regression |
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Probability (3 weeks) |
Proper and Improper Integrals |
Discrete and Continuous Distributions Reliability Theory |
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Inferential Statistics8,10 (1.5 weeks) |
Quality Control Confidence Intervals Simulations |
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8 Lake Pollution Project (Parts 1 & 2, then Part 4, then Parts 5 & 6) 9 Project on Web-Data Collection 10 Simulation |
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