Summer 2015

Calculus: Single Variable (coursera)

Functions

Differentiation

  • Derivatives -Definition and interpretations of the derivative
  • Differentiation rules -Rules for differentiating combinations of functions
  • Linearization - First order Taylor approximations
  • Higher derivatives -Definition and interpretation of higher derivatives
  • Optimization -Classifying critical points and finding extrema
  • Differentials -Implicit differentiation and related rates
  • Differentiation as an operator -Using operators to compute other derivatives

Integration

  • Antidifferentiation -The indefinite integral and separable differential equations
  • Exponential growth examples -More examples of exponential growth and decay
  • More differential equations -Linear first order differential equations
  • ODE Linearization -Solving harder differential equations
  • Integration by Substitution -Substitution as an integration technique
  • Integration by parts -Using the product rule as an integration technique
  • Trigonometric substitution -Integration using trigonometric substitutions
  • Partial fractions -Integration of rational functions using algebra
  • Definite integrals -Definition and interpretation of the definite integral
  • Fundamental Theorem of Integral Calculus -Connecting definite and indefinite integrals
  • Improper integrals -Computing definite integrals when FTIC does not apply
  • Trigonometric integrals -Products and powers of trigonometric functions
  • Tables and computers -Using tables of integrals and mathematics software

Applications

  • Simple Areas -Finding the area of regions in the plane
  • Complex Areas -Areas of more complex regions in the plane
  • Volumes -Using the volume element to compute volume
  • Volumes of revolution -Volumes from revolving a region about an axis
  • Volumes in arbitrary dimension -Fourth dimension and beyond
  • Arclength -Finding the length along a curve
  • Surface area -Surface area of a solid of revolution
  • Work -Computing work with integration
  • Elements -Pressure, force, and other applications
  • Averages -The average value of a function over an interval
  • Centroids and centers of mass -Finding centroid and center of mass with integration
  • Moments and gyrations -Moment of inertia and radius of gyration
  • Fair probability -Uniform distribution
  • Probability densities -Using the density function to compute probabilities
  • Expectation and variance -Properties and interpretations of probability distributions

Discretization

  • Sequences -Discrete-input functions
  • Differences -Derivatives of sequences
  • Discrete Calculus -How to integrate discrete functions
  • Numerical ODEs -Using sequences to solve ODEs
  • Numerical integration -Using sequences to solve definite integrals
  • Series -Infinite series as improper discrete integrals
  • Convergence Tests -Comparison-type tests
  • Convergence tests 2 -Geometric series-type tests
  • Absolute Convergence and Conditional Convergence -Two types of series convergence
  • Power Series -Interval and radius of convergence
  • Taylor series redux -Details about Taylor series convergence
  • Approximation and error -How to estimate an infinite series


Links

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