1
Precalculus Review

Notes on Transcendental Functions
Graphs of Functions You Should Know

Exercises - Review of Some Ideas from Precalculus
2
Introduction to Calculus
Tangent Lines, Velocity, and Other Rates
An Intuitive Way to Think About Limits
The Epsilon Delta Definition of a Limit
Examples of Using the Epsilon-Delta Definition
Exercises - Tangent Lines, Velocity, and Other Rates
Exercises - The Epsilon-Delta Def. of a Limit (and Review)
3
Limits
Proof of the Limit of a Sum Law
The Limit Laws

Infinite Limits
Limits at Infinity
Exercises - Limits

4
More Limits
Comparisons of Functions
Limits of Compositions
Exercises - Comparisons of Functions (and Review)
Exercises - Limits of Compositions (and Review)

More Practice - Limit Laws
More Practice - Infinite Limits and Limits at Infinity
More Practice - Limits
5
Continuity
Continuous Functions and Discontinuities
Exercises - Continuous Functions (and Review)

More Practice - Continuity
6
The Intermediate Value Theorem
The Intermediate Value Theorem (IVT)
Proof of the IVT (for the curious)
Exercises - Intermediate Value Theorem (and Review)

More Practice - Intermediate Value Theorem
7
Derivatives and Differentiability
The Definition of the Derivative
Differentiability
Acceleration, Velocity, and Speed
Exercises - Definition of Derivative (and Review)
Exercises - Differentiability

More Practice - Definition of Derivative
More Practice - Differentiability and Continuity
8
Induction
Summations and Sigma Notation
Two Important Properties of Sums (Linearity)
Induction
More Exercises - Induction and Sums
9
Basic Rules of Differentiation
Basic Derivative Rules
Proof Derivatives of Constant Functions are Zero
The Binomial Theorem
Proof of the Power Rule (for positive integer powers)
Proof of the Derivative of the Sine Function
Proof of the Sum and Difference Rules
Proof of the Constant Multiple Rule
Proof of the Product Rule
Proof of the Quotient Rule
Exercises - Basic Derivative Rules (and Review)
Exercises - Derivatives Involving Trig (and Review)
10
Chain Rule
Proof of the Chain Rule (for Compositions)
Exercises - The Chain Rule (and Review)

More Practice: The Chain Rule
12
Implicit Differentiation
Implicit Differentiation
The Power Rule for Derivatives (for rational powers)
Exercises - Implicit Differentiation (and Review)

More Practice - Implicit Differentiation
11
More Derivative Rules
Inverse Trigonometric Functions
Derivatives of Inverse Trigonometric Functions

A Natural Question
Logarithmic Differentiation
Exercises - More Derivative Rules

Exercises - Higher Order Derivatives
Exercises - Logarithmic Differentiation
Exercises - Finding Derivatives (Mixed Techniques)
12
Related Rates    Exercises - Related Rates

More Practice - Related Rates
13
Differentials and Approximation    Exercises - Differentials and Approximation
14
Extrema
Extrema of Functions
Proof of the Boundedness Theorem (for the curious)
Proof of the Extreme Value Theorem (for the curious)
Proof of Fermat's Theorem
Exercises - Extrema of Functions
15
Mean Value Theorem
Proof of Rolle's Theorem
Proof of the Mean Value Theorem
Proof that Functions with Zero Derivatives are Constant
Exercises - The Mean Value Theorem

More Practice - The Mean Value Theorem
16
Graphing
The First Derivative Test
The Second Derivative Test
Concavity
The Graphing Handout

Exercises - Graphing Functions
(Note, this set of exercises duplicates many of the problems
in the Graphing Handout - it's a work in progress)
17
Optimization    Exercises - Optimization
19
Antiderivatives
U-Substitution
Exercises - Antiderivatives and U-Substitution
20
Separable Differential Equations    Exercises - Separable Differential Equations
21
Riemann Sums and the Definite Integral
Summation Techniques
Approximating the Area Under a Function
Riemann Sums and the Definite Integral
Finding Definite Integrals Directly from Riemann Sums
Exercises - Summation Techniques
Exercises - Riemann Sums
21
The Average Value of a Function
The Mean Value Theorem for Integrals
Exercises - Mean Value Theorem for Integrals
22
The Fundamental Theorem of Calculus
Evaluating Definite Integrals with Antiderivatives
Properties of the Definite Integral
The Derivative of a Definite Integral Function
Exercises - The Fundamental Theorem of Calculus (Part I)
Exercises - Fundamental Theorem of Calculus (Part II)
23
More on U-Substitution    Exercises - More on U-Substitution
24
Area between Curves    Exercises - Area Between Curves