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Calculus Lecture Notes

Eric S. Key

Contents
An introduction to calculus
- Constant functions
- Linear functions
- Factoring differences of powers
- Squaring functions
- Cubing functions
- The general case
- The fundamental theorem of calculus
The rates of change of some other functions
Some extensions of our results
- Positive rational powers
More on rates of change
- Rates of change of sums
- Two applications
What do we mean by ``approaching''?
The funnelling theorem:
Formalizing the concept of ``approaching''
- Constant functions:
- Linear functions
- Quadratic functions
- The sum of two functions
- Other properties
- An abuse of notation and terminology
Homework, due 1/28:
Infinity
Continuity
More rates of change for familiar functions
- The tangent function
- Negative integer powers
- Rates of change for inverse functions
- The natural logarithm
- The arctangent function:
- The arcsine function:
Another extension of the power rule
- The algebraic computation
- The geometric approach
- Some gaps in our reasoning
Information from derivatives
Product rule, quotient rule, and chain rule
- Differentiating products:
- Differentiating quotients
- Composite functions:
More on composite functions
- Continuity composite functions
- Differentiabilty and continuous difference quotients
- Another application of the same idea
Summary of differentiation
- The definition
- The interpretation
- General rules
- Specific derivatives:
- Additional derivatives:
- A note on the power rule
Some simple applications of differentiation
- Growth and Decay Problems
- Hooke's Law
- Implicit Differentiation
- Related rates
- Numerical solutions of equations
Homework, due 2/23
Two important properties of functions
- The range of a continuous function
- Where maxima are not located
- Applications of Rolle's Theorem
  - The mean value theorem
Some examples of searching for extreme values
The fundamental theorem of calculus
More on limits
- Related facts about the natural logarithm
- Cauchy's Mean Value Theorem and L'Hopital's Rule
  - Examples
- Polynomial Approximations
  - Approximating sine for small values
  - Approximating the natural logarithm
Linear approximations
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