Techniques of Calculus, I

Ilustration of a geometric method of determining the volume of a torus.

Credit: Image adapted from a figure in Applied Calculus: Principles and Applications by Robert Gibbes Thomas, public domain

Resource Description

A Jupyter notebook companion for a first course in Calculus, including a review of Precalculus concepts.

Course Number

MATH 110

License

MIT

Resource
Author(s)
Table of Contents

Online Resource

View the entire resource online here: Techniques of Calculus, I

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Jan Reimann

Associate Professor
Department of Mathematics
Penn State University
318B McAllister
University Park, PA 16802

Email: jan.reimann@psu.edu

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Precalculus Review
- Algebraic Simplification Techniques
- Factoring Techniques
- Solving Equations
- Graphing
- Solving Inequalities
- The Domain of a Function
- Try It Yourself
Functions
- Functions, Domain, and Range
- Composition of Functions
- Cost, Revenue, and Profit
- Power, Polynomial and Rational Functions
- Demand and Supply
- An Application
- Try It Yourself
Limits and Continuity
- Limit of a Function
- Properties of Finite Limits
- Examples: Computing Limits from Graphs
- Examples: Computing Limits Algebraically
- Indeterminate Forms
- Limits at Infinity
- Continuity
- Try It Yourself
Derivatives
- The Concept of the Derivative
- Basic Rules of Differentiation
- Try It Yourself
Marginal Analysis
- Cost, Revenue, and Profit
- Marginal Analysis
- Elasticity of Demand
- Elasticity & Total Revenue
- Try It Yourself
Implicit Differentiation
- Implicit Differentiation
- Related Rates
- Try It Yourself
Curve Sketching
- Intervals of Increase and Decrease
- Relative Extrema
- Critical Points
- The First Derivative Test
- The Second Derivative Test
- Intervals of Concavity
- Inflection Points
- Asymptotes
- Curve Sketching
- Try It Yourself
Optimization
- Relative Extrema
- Critical Points
- The First Derivative Test
- The Second Derivative Test
- Absolute Extrema
- Applications
- Try It Yourself
Exponential and Logarithmic Functions
- Exponential and Logarithmic Functions
- Laws of Exponential and Logarithmic Functions
- Logarithmic Differentiation
- Try It Yourself
Compound Interest
- Notation and Terminology
- Accumulated Amount Examples
- Present Value Examples
- Try It Yourself
Indefinite Integrals
- Indefinite Integrals
- Basic Rules of Integration
- Initial Value Problem
- Integration by Substitution
- Try It Yourself
Definite Integrals
- Riemann Sums and the Definite Integral
- The Fundamental Theorem of Calculus
- Substitution for Definite Integrals
- Average Value of a Function
- Try It Yourself
Applications of Definite Integrals
- Area Between Two Curves
- Consumer, Producer, and Total Surplus
- Income Streams and Annuities
Integration by Parts and Improper Integrals
- Integration by Parts
- Application of Integration by Parts
- Improper Integrals
- Application of Improper Integrals