Thread: Math 1241 Topics covered in the course Math 1241, Calculus and Dynamical Systems in Biology at the University of Minnesota. Background material Functions The function machine Function notation Function examples Function machine parameters The function machine inverse Inverse function examples Function composition Function machine composition Function composition examples Functions and inequalities worksheet Exponentials and logarithms Basic rules for exponentiation Basic idea and rules for logarithms The exponential function Redundant parameters in the exponential function Modeling Data modeling versus mathematical modeling Discrete dynamical systems Dynamical system introduction The idea of a dynamical system An introduction to discrete dynamical systems Discrete dynamical systems worksheet 1 Some initial models An infectious disease model Introduction to an infectious disease model A discrete SIR infectious disease model Controlling a rabbit population Controlling a rabbit population Equilibria in discrete dynamical systems Linear models Initial bacteria growth model Developing an initial model to describe bacteria growth Bacteria growth model exercises Bacteria and equilibria worksheet Penicillin clearance model Constructing a mathematical model for penicillin clearance Penicillin clearance model exercises Worksheet for constructing a mathematical model for penicillin clearance Exponential growth and decay Exponential growth and decay modeled by discrete dynamical systems Discrete exponential growth and decay exercises Doubling time and half-life of exponential growth and decay Chemical pollution A model of chemical pollution in a lake Chemical pollution model exercises Worksheet: Chemical pollution worksheet Cobwebbing and stability Using cobwebbing as a graphical solution technique for discrete dynamical systems The idea of stability of equilibria for discrete dynamical systems Problems Elementary discrete dynamical systems problems Elementary discrete dynamical systems biology problems Solutions to elementary discrete dynamical systems problems Solutions to elementary discrete dynamical systems biology problems Review questions: Elementary discrete dynamical system problems Review questions: Elementary discrete dynamical system biology problems Study guide for Math 1241 discrete dynamical system module exam The derivative Preliminary exploration Determining stability by cobwebbing linear approximations around equilibria Worksheet on determining stability by cobwebbing linear approximations around equilibria Introduction to the derivative Approximating a nonlinear function by a linear function Developing intuition about the derivative Derivatives of polynomials Calculating the derivative of a linear function using the derivative formula Calculating the derivative of a quadratic function The derivative of a power function Derivatives of polynomials Derivatives of exponentials and logarithms Exploring the derivative of the exponential function The derivative of the natural logarithm Differentiation rules The idea of the product rule The quotient rule for differentiation The idea of the chain rule Simple examples of using the chain rule The derivative and graphing Partial derivatives Introduction to partial derivatives Partial derivative examples Problems Review questions: Elementary derivative problems Review questions: Elementary partial derivative problems Study guide for Math 1241 differentiation module exam Applications of differentiation Stability of equilibria in discrete dynamical systems The stability of equilibria for discrete dynamical systems Examples of determining the stability of equilibria for discrete dynamical systems Logistic growth Environmental carrying capacity Developing a logistic model to describe bacteria growth Worksheet: Worksheet on developing a logistic model to describe bacteria growth Minimization and maximization Maximization and minimization Harvest of natural populations Harvest of natural populations exercises Taylor polynomials The tangent line as a linear approximation Taylor polynomial introduction Quadratic Taylor polynomial examples Approximating functions by quadratic polynomials Problems Elementary discrete dynamical systems problems, part 2 Elementary discrete dynamical systems biology problems, part 2 Minimization and maximization problems Linear and quadratic Taylor polynomial problems Solutions to elementary discrete dynamical systems problems, part 2 Solutions to elementary discrete dynamical systems biology problems, part 2 Solutions to minimization and maximization problems Solutions to linear and quadratic Taylor polynomial problems Study guide for Math 1241 applications of differentiation module exam Review questions: Elementary discrete dynamical systems biology problems, part 2 Autonomous differential equations Introduction From discrete dynamical systems to continuous dynamical systems Worksheet on going from discrete dynamical systems to continuous dynamical systems Introduction to autonomous differential equations Solution methods Solution to linear equations Simple linear differential equations Graphical methods Solving single autonomous differential equations using graphical methods Worksheet: Solving single autonomous differential equations using graphical methods Stability of equilibria Exponential growth and decay modeled by continuous dynamical systems Worksheet: Exponential growth and decay The stability of equilibria of a differential equation Worksheet: Stability of equilibria of a differential equation Numerical solution The Forward Euler algorithm for solving an autonomous differential equation Worksheet: Forward Euler introduction Guided practice: Forward Euler and linear approximations Bifurcations Introduction to bifurcations of a differential equation Worksheet: Introduction to a bifurcation diagram Spruce budworm outbreak model An infectious disease model Model of an infectious disease without immunity Worksheet: Model of an infectious disease without immunity Problems Review questions: Single autonomous differential equation problems Study guide for Math 1241 autonomous differential equation module exam Two dimensional autonomous differential equations Biological models An infectious disease model An infectious disease model with immunity Worksheet: Sketch evolution of an SIR model Dynamics of competition The dynamics of competition between two species Worksheet: The dynamics of competition Predator-prey dynamics Introducing rabbit predators A spiking neuron A simple spiking neuron model Phase plane analysis Introduction to visualizing differential equation solutions in the phase plane Problems Review questions: Two dimensional autonomous differential equation problems Study guide for Math 1241 final exam Part 17 Elementary derivative problems Elementary partial derivative problems Solutions to elementary derivative problems Solutions to elementary partial derivative problems Integration Elementary integral problems Solutions to elementary integral problems Developing intuition about the indefinite integral Using the Forward Euler algorithm to solve pure-time differential equations Calculating the area under a curve using Riemann sums