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Calculus Single Variable

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Introduction

  1. Introduction - Prerequisites and general info

Functions

  1. Functions - Definition and examples of functions
  2. The Exponential - The exponential function defined
  3. Taylor series - The Taylor series defined and applied
  4. Computing Taylor series - Using composition to compute Taylor series
  5. Convergence - Problems with some Taylor series
  6. Expansion points - Taylor series expansions about other points
  7. Limits - Definition of limit and continuity
  8. L'Hopital's Rule - Statement and examples in a Taylor series context
  9. Orders of growth - Relative growth of the most common functions

Differentiation

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

Integration

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

Applications

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

Discretization

  1. Sequences -Discrete-input functions
  2. Differences -Derivatives of sequences
  3. Discrete calculus -How to integrate discrete functions
  4. Numerical ODEs -Using sequences to solve ODEs
  5. Numerical integration -Using sequences to solve definite integrals
  6. Series -Infinite series as improper discrete integrals
  7. Convergence tests 1 -Comparison-type tests
  8. Convergence tests 2 -Geometric series-type tests
  9. Absolute and conditional -Two types of series convergence
  10. Power series -Interval and radius of convergence
  11. Taylor series redux -Details about Taylor series convergence
  12. Approximation and error -How to estimate an infinite series
  13. Calculus -What we can and cannot do...for now

Wrap-Up

  1. Foreshadowing -Towards multivariable calculus