A Brief Introduction to Calculus
1
,Contents
Introduction
1. Functions and Graphs
2. Linear Functions, Lines, and Linear Equations
3. Limits
4. Continuity
5. Linear Approximation
6. Introduction to the Derivative
7. Product, Quotient, and Chain Rules
8. Derivatives and Rates
9. Increasing and Decreasing Functions
10. Concavity
11. Optimization
12. Exponential and Logarithmic Functions
13. Antiderivatives
14. Integrals
2
,Introduction
These notes are intended as a brief introduction to some of the main
ideas and methods of calculus. They are very brief and are not
intended as a mathematical exposition of the subject. They do not
contain recipes for solving problems. Hence, you will not be able to
solve homework problems by looking back through the notes and
finding similar examples.
We feel that the only way one can really learn calculus (or any another
subject) is to take basic ideas and apply those ideas to solve new
problems. Hence the learning process is accomplished primarily by
solving the problems.
These notes were prepared with support from a National Science
Foundation Grant.
Copyright: Robert Molzon
Copies of these notes may be made under the terms of the General
Public License.
3
, 1. Functions and Graphs
A function is a rule that assigns one number to a given number. In
general “number” will mean real number such as 1.25 or 6.498, 2 , or
π. The rule that defines the function can be described in several
different ways. Perhaps the most common way of describing the
function is an algebraic expression. For example
ft = t 2
is the function that assigns the square of number to any given number.
The notation above is sometimes referred to as “functional notation”.
We may compute the value the function assigns to a given number by
substitution the given number into the algebraic expression and then
performing the algebra. For example, with the above function ft = t 2
we evaluate
f2 = 2 2 = 4
f3 = 3 2 = 9
f4.1 = 4.1 2 = 16.81
Functions may also be described by tables or lists. Suppose we
measure some quantity such as weight or length at certain fixed times
and express the results in a table. For example, suppose our
measurements give the following results.
Time Weight
1 3.2
2 4.5
3 6.2
4 7.8
5 9.3
We can then consider the table as a rule that assigns a number
(weight) to a given time. Notice that this function is only defined for the
values of time 1, 2, 3, 4, and 5. If we denote the function described by
the table using functional notation we could write Wt to denote the
weight at time t. In this case Wt only makes sense for the t values
above.
4
1
,Contents
Introduction
1. Functions and Graphs
2. Linear Functions, Lines, and Linear Equations
3. Limits
4. Continuity
5. Linear Approximation
6. Introduction to the Derivative
7. Product, Quotient, and Chain Rules
8. Derivatives and Rates
9. Increasing and Decreasing Functions
10. Concavity
11. Optimization
12. Exponential and Logarithmic Functions
13. Antiderivatives
14. Integrals
2
,Introduction
These notes are intended as a brief introduction to some of the main
ideas and methods of calculus. They are very brief and are not
intended as a mathematical exposition of the subject. They do not
contain recipes for solving problems. Hence, you will not be able to
solve homework problems by looking back through the notes and
finding similar examples.
We feel that the only way one can really learn calculus (or any another
subject) is to take basic ideas and apply those ideas to solve new
problems. Hence the learning process is accomplished primarily by
solving the problems.
These notes were prepared with support from a National Science
Foundation Grant.
Copyright: Robert Molzon
Copies of these notes may be made under the terms of the General
Public License.
3
, 1. Functions and Graphs
A function is a rule that assigns one number to a given number. In
general “number” will mean real number such as 1.25 or 6.498, 2 , or
π. The rule that defines the function can be described in several
different ways. Perhaps the most common way of describing the
function is an algebraic expression. For example
ft = t 2
is the function that assigns the square of number to any given number.
The notation above is sometimes referred to as “functional notation”.
We may compute the value the function assigns to a given number by
substitution the given number into the algebraic expression and then
performing the algebra. For example, with the above function ft = t 2
we evaluate
f2 = 2 2 = 4
f3 = 3 2 = 9
f4.1 = 4.1 2 = 16.81
Functions may also be described by tables or lists. Suppose we
measure some quantity such as weight or length at certain fixed times
and express the results in a table. For example, suppose our
measurements give the following results.
Time Weight
1 3.2
2 4.5
3 6.2
4 7.8
5 9.3
We can then consider the table as a rule that assigns a number
(weight) to a given time. Notice that this function is only defined for the
values of time 1, 2, 3, 4, and 5. If we denote the function described by
the table using functional notation we could write Wt to denote the
weight at time t. In this case Wt only makes sense for the t values
above.
4
