These are notes for 1st semester calculus - Limits, derivatives, applications of derivatives, basic integration.

*Comments:* The section on limit theorems contains a lot of epsilon-delta proofs, which are usually not covered in a first-term calculus course. The related rates examples are complete, but need reworking. I need to show a graphing example step-by-step, the way I would do one in class. I need to write up proofs of L'Hôpital's rule and the "only one critical point" theorem that is used in the max-min word problems section; this will require sections on the extended Mean Value Theorem and the Intermediate Value Theorem for derivatives. The extended Mean Value Theorem is usually proved using Rolle's Theorem, which is proved using the existence of maxima and minima for a continuous function on a closed interval. That result in turn requires foundational stuff; you can use completeness for the reals, the Nested Interval Theorem, or Dedekind cuts [which are all equivalent]. That's a lot of writing! So this is my excuse for not having done it yet.

I need to write up the section on hyperbolic functions.

[May 19, 2024] I corrected a couple of typos in the notes on limit theorems.

- Introduction to limits:
informal computations, graphical and numerical evidence.

[PDF file] - The epsilon-delta definition of a limit

[PDF file] - Properties of limits (no proofs)

[PDF file] - Limit theorems - proofs of many of the theorems on limits (
*theory - not required for first-term calculus*)

[PDF file] - Left and right-hand limits; infinite limits

[PDF file] - Continuity; the Intermediate Value Theorem

[PDF file] - Limits at infinity; horizontal and vertical asymptotes

[PDF file] - Derivatives, tangent lines, and rates of change

[PDF file] - Differentiation rules (through the Product Rule and Quotient Rule)

[PDF file] - The Chain Rule

[PDF file] - Trig limits and derivatives

[PDF file] - Inverse
functions and the derivative of an inverse

[PDF file] - Derivatives of log and exponential functions

[PDF file] - Implicit differentiation

[PDF file] - Related rates

[PDF file] - The Mean Value Theorem

[PDF file] - Differentials

[PDF file] - Newton's method

[PDF file] - Absolute maxima and minima

[PDF file] - Increasing and decreasing functions

[PDF file] - Concavity and the Second Derivative Test

[PDF file] - Graphing curves

[PDF file] - Max-min word problems

[PDF file] - Antiderivatives

[PDF file] - Substitution

[PDF file] - Sums and summation notation

[PDF file] - Rectangle sums

[PDF file] - Definite integrals

[PDF file] - The Fundamental Theorem of Calculus

[PDF file] - Area between curves
(see the area notes for 2nd-semester calculus as well)

[PDF file] - L'Hopital's Rule

[PDF file] - Calculus of the natural logarithm; logarithmic differentiation

[PDF file] - Separation of variables (brief introduction)

[PDF file] - Exponential growth and Newton's law of cooling

[PDF file] - Inverse trig functions

[PDF file]

Copyright 2020 by Bruce Ikenaga