calculus.seas.upenn.eduPennCalc | Main / Calculus Single Variable

Recent Changes - Search : View Edit History Print Main / Calculus Single Variable About the PennCalcWiki - PennCalc SVC Table of Contents Introduction Functions Differentiation Integration Applications Discretization 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 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 Introduction loop trail Edit - History - Print - Recent Changes - Search Page last modified on August 25, 2018, at 04:13...