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# Calculus Single Variable

## PennCalc SVC

UPDATE: August 2018: The PennCalcWiki is broken, thanks to a server upgrade to PFP-7 which renders the skins and latex plugin unusable. Sadly, this wiki cannot easily be converted to be PHP-7 compatible. You might want to try the wayback machine.

## 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

Page last modified on August 25, 2018, at 04:13 PM