Lecture 5 - the First Big Problem of Calculus

Lecture 5 - The First Big Problem of Calculus

-> The first big problem of Calculus is the tangent line problem, or, given a function f(x), we would like to find the slope of the line tangent ('touching') to f at x = c. A picture's worth a thousand words:

For example, we would like to find mtan, the slope of the above tangent line, at c = 1.

-> The problem arises when we recall how we found slopes before. The formula is

where we have two known points on the line, and . In the tangent line problem, only one point is known - (c, f(c)). So we have to take a little more sophisticated approach.

If we know any two points on the line, then we are able to find the slope of what we call a secant line to the function at those two points.

If the second point is an arbitrary distance, which we denote , down the x-axis from c, then it's coordinates are (c, f(c + )), and the slope of the secant line containing the 2 points is msec = . Now, here comes the tricky part - what we can do is take a look at the secant lines as we repeat this process, with becoming smaller every time:

Take a look at the pictures - as gets smaller, the secant lines begin to resemble the tangent line more and more - we say that they 'approach' the tangent line. And so, as gets smaller, msec gets closer and closer to mtan. Now what we need is some way to 'hard-code' this realization that we've made from looking at the pictures, that is, we need to turn it into a formula:

slope "f prime "as "mtan approaches

of at c" gets small" msec"

tangent

-> For any function f(x), we have a formula, known as the general derivative of f (or just 'derivative'), which can give the slope of the tangent at any point along f . It is the solution to the 1st big problem of Calculus, and we define it:

.

There is alternative notation as well:

Ex: For each function, find and the slope of the function at x = 2.

(a)

(b)

(c)

(2) Show that f is not differentiable (i.e. f ' does not exist), at x = 0 for the function . Discuss the connection between continuity and differentiability.

(3) Find f' for f(x) = mx + b

This last Example (and to some extent,1(b) and 1(c) as well), point the way to a new approach to the important problem of finding derivatives - maybe we can sidestep the 'limit process' by formulating rules for certain 'standard functions' . And in fact, in subsequent sections, that is exactly what we will do.