Calculus Study Resources

Calculus

The following introductory single-variable calculus study resources were developed by:
Kayla Jacobs
Technion International School of Engineering
104003 Differential and Integral Calculus I
Fall 2010 and Fall 2011
Course lecturer: Ross Pinsky

Week	Topics	Resources
1	Welcome, Bounds, and Finite Limits Introduction Bounds Upper and lower bounds Infimum (greatest lower bound) Supremum (least upper bound) Definition of finite limit for sequences	Handouts Summary Reference: Math-English Dictionary Strategy for Finite Limit Problems Finite Limit Problems with Solutions Recommended Supplementary Calculus Books Links The Calculus Page Problems List - By D.A. Kouba. A great set of many calculus problems to practice, with full solutions.
2	More on Limits Limit convergence for sequences Helpful limits to know by heart Algebra of limits Limits of polynomial quotients Sandwich Lemma	Handouts Summary Links Suggested Guidelines for Writing Proofs - By John M. Lee Humorous Proof "Techniques" - DON'T use these! :) Review: Solving Quadratic Inequalities
3	Functions and their Limits: Domain, image, and range of a function Increasing and decreasing functions Injective, surjective, and bijective functions (1-to-1, onto, and both) Inverse functions Elementary operations and elementary functions Definition of a finite limit of a function as x->a Algebra of limits for functions	Handouts Summary
4	Even More Limits, and Continuity More limits: One-sided limits Finite limit as x->infinity "Infinite" limit as x->a "Infinite" limit as x->infinity Continuity: Definition of continuity Algebra of continuity Continuous functions	Handouts Summary
5	More Continuity, and Intermediate Value Theorem (IVT) One-sided continuity Intermediate Value Theorem (IVT) IVT application: Root-finding	Handouts Summary
6	Discontinuities and Derivatives Discontinuities Removable discontinuity Discontinuity of the 1st kind ("jump") Discontinuity of the 2nd kind Derivatives Definition Intuitive definitions Practical examples General differention rules Derivatives of simple functions Derivatives of exponential and logarithmic functions Derivatives of simple functions Derivatives of trigonometric functions Derivatives of hyperbolic functions	Handouts Summary Reference: Delightful Differentiation
7	More Derivatives Derivative rules Review: Basic trigonometric identities Review: Logarithms and exponentials	Handouts Worksheet: Derivative Practice Reference: Review: Basic Trigonometric Identities, Logarithms and Exponentials
8	Mean Value Theorem (MVT), Linear Approximations, and L'Hopital's Rule Mean Value Theorem (MVT) Rolle's Theorem (special case of MVT) MVT (= Lagrange's Theorem) MVT applications Cauchy's Theorem (generalization of MVT) Linear Approximations L'Hopital's Rule	Handouts Summary
9	Taylor Series and Max-Min Problems Taylor series Taylor polynomials Maclaurin polynomials Maclaurin polynomial example: sin(x) DeMoivre's Theorem Max-min problem-solving strategy	Handouts Summary
10	Convexity and Integration Convexity Inflection point Indefinite integrals Integration by parts Integration of rational functions using partial fractions	Handouts Summary Links Wolfram Integrator Step-By-Step Partial Fractions Decomposer - By Sam Blake
11	Integrals Indefinite vs. definite integrals Fundamental Theorem of Calculus Definite integral properties Area in [a,b] bounded by curve Average value Riemann sum Integration method: u-substitution Basic trigonometric derivatives and indefinite integrals Integration method: Trigonometric substitution for rational functions of sines and cosines Integration method: Trigonometric substitution	Handouts Summary
12	Improper Integrals and Infinite Series Improper integrals Definition Solving strategy Comparison Theorem p-Test Infinite series Partial sums for series Convergence tests Choosing a convergence test	Handouts Summary Reference: Infinite Series Convergence Tests Worksheet: Infinite Series Convergence Practice Links Partial Sums Example (see page 3) - By Amy Austin Flowchart: Choosing a Convergence Test - By David J. Manuel