See
handout for administrative information.
We will cover three main topics this quarter. Real numbers, limits,
and derivatives. We will begin with a review of real numbers and move
quickly into the algebra of inequalities. Then we will cover the highlight
of the quarter: epsilon-delta proofs of limits. Once we have this down,
it will be easy to give a rigorous understanding of continuous functions.
And of course, it would not be calculus if we did not study derivatives
of the calculus kind.
If Amy has three more children, then she will have more than five children. How many children does she have?
She has more than two children. Compare this question with
the algebraic question,
`a+3=5. What is a?' To solve the
equality, you would subtract three from both sides. That is also
how you would solve the inequality.
A limit statement contains a number and a function. It is a statement about the values of the function near the number, and has absolutely nothing to do with the value of the function at the number. Loosely stated, the limit of a function f as the function approaches the number c is the value that the function f approaches when evaluated using numbers near c. The limit only exists if there is an unambiguous choice. We will see how the epsilon-delta definition of limits is used to find out when the choice of a limit is unambiguous.
A function is continuous if the limit agrees with the value
of the function every time. This agrees with the notion that
it should have no breaks or jumps in its graph. Going back to
limits, this gives a decent picture of what a limit is. A limit
is the value that the function `should be.' Note that the function
f(x)=1/x is a continuous function since it is not defined at zero.
It is continuous everywhere it is defined.
A function, if it is nice enough, can be approximated well by a straight line, usually called the tangent line. The slope of this line is called the derivative. We will see that it is easy to define it using the limit, because the tangent line is the limit of other lines. Hence the slope of the tangent line is the limit of the slope of the approximating lines. This is a very good use of the limit since the apporximating lines can be drawn concretely. The tangent line is more theoretical and can only be drawn after one has the answer!
Naturally, we will do many concrete examples of finding the derivative. Since the derivative of a function depends on the point for which the tangent line is drawn, the derivative can be thought of as a function. The derivative of f(x)=x is f'(x)=1. The derivative of f(x)=x^2 , i.e. x squared, is f'(x)=2x. The derivative of f(x)=x^3, i.e. x to the third power, is f'(x)=3x^2. For f(x)=x^4, it is f'(x)=4x^3. Do you see the pattern?
For monomials of the form f(x)=x^n, the derivative function
is f'(x)=nx^(n-1). This use of n means that this is true
for any integer n larger than zero. This is a common way
for us to write general formulas which we will apply to special
cases over and over again. Of course, we will do derivatives
for other functions as well, but these are the easiest.
This is just an introduction to the many exciting topics
we will cover this quarter. Next quarter, we will move on
to applications of the derivative to real life problems, and
we will learn about the integral. What better time, than
late spring to discover the fundamental theorem of calculus?