Exam I

• Modeling
  • Differential Equations
  • Seperation of Variables
  • Euler's Method
  • Population Models
    • Exponential Growth
    • Logistic Growth
    • The Coalition Model
• Methods of integration
  • substitution
  • integration by parts
  • using tables
  • partial fractions
• The First and Second Fundamental Theorems of Calculus
• Definite Integrals, Area, Average and Total Change of Functions
• Approximating Definite Integrals
  • Right and Left Riemann Sums

Exam II

• Approximating Definite Integrals
  • The Midpoint Rule
  • The Trapezoid Rule
  • Simpson's Rule
• Improper Integrals
  • limits going to infinity
  • limits to points where integrand is not defined
• Volumes by slicing
• Work

• Probability
  • Events
  • Random Variables
  • Expected Values
• Series
  • partial sums, geometric series
  • the n^th term test
    • the harmonic series
    • the sum of x^-p for 0<p<=1 and p>1
  • convergence tests
    • the comparison test
    • the ratio test
    • the integral test

Exam III

• Probability and Distribution Functions
  • density functions
    • mean, median
  • cumulative probability distribution functions
  • normal distributions
    • mean, standard deviation
• Taylor Approximations
  • linear approximation
  • second, third, n^th order Taylor polynomials
  • convergence of Taylor series

• Series, part II
  • Alternating Series; The Alternating Series Test
  • Absolute Convergence
  • The Extended Ratio Test
  • Convergence of Power Series
  • Taylor Series and intervals of convergence
  • Fourier Series
• Oscillations
  • Complex numbers
    • real and imaginary parts
    • polar notation
    • Euler's identity
  • second order differential equations
    • simple harmonic motion s''+(k/m)s=0
    • damped harmonic motion s''+(a/m)s'+(k/m)s=0
    • general solutions to second order linear differential equations with constant coefficients
    • characteristic equations
  • Series Solutions to Differential Equations
    • solutions expressed as Taylor series about a point
  • predator-prey model and phase planes
    • systems of first order differential equations
    • phase planes
    • trajectories
    • equilibrium points
  • SIR model and phase planes
  • phase plane analysis
    • nullclines
    • analyzing trajectories in regions defined by nullclines
    • equilibrium points and nullclines
    • applications to the SIR model