
PennCalc SVC
Table of Contents
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 Discreteinput 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 Comparisontype tests
 Convergence tests 2 Geometric seriestype 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
WrapUp
 Foreshadowing Towards multivariable calculus
