PennCalc SVC
Table of Contents
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Introduction
- Introduction - Prerequisites and general info
Functions
- Functions - Definition and examples of functions
- The Exponential - The exponential function defined
- Taylor series - The Taylor series defined and applied
- Computing Taylor series - Using composition to compute Taylor series
- Convergence - Problems with some Taylor series
- Expansion points - Taylor series expansions about other points
- Limits - Definition of limit and continuity
- L'Hopital's Rule - Statement and examples in a Taylor series context
- Orders of growth - Relative growth of the most common 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 1 -Comparison-type tests
- Convergence tests 2 -Geometric series-type tests
- Absolute and conditional -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
- Calculus -What we can and cannot do...for now
Wrap-Up
- Foreshadowing -Towards multivariable calculus