At the start of today's discussion we are going to consider two fairly general problems. As we work on solving those problems, we will begin to notice some similarities between them.
A line which is tangent to a curve at some point just touches the curve without crossing it. Before the invention of calculus, geometers knew that most curves had unique tangents at every point on the curve. In many cases they even developed clever geometric construction techniques for constructing the tangent lines for many different kinds of curves.
None of these construction techniques had anything to do with the concept of a limit. In every case the construction was done using simple tools such as a straightedge and a compass. Here is an example to give you the flavor of these methods.
The geometric technique for constructing a parabola is to draw a line, called the directrix, and a point not on the line, called the focus. The set of all points in space that lie an equal distance from the focus and the directrix traces out a curve called a parabola. For example, the parabola with focus at (0,4) and directrix equal to the line y = -4 has equation y = x2/16.
Consider the green line that connects the focus (0,4) to the point where the vertical blue line touches the directrix at (r,-4). If you draw a line passing through the midpoint of the green line and the point where the two blue lines meet, you have constructed a tangent line to the parabola.
The problem with these techniques for constructing tangent lines is that they required considerable cleverness to come up with and they also were highly specific to the curves. The technique that one used to construct a tangent for a parabola was very different from the technique used to construct the tangent to an ellipse, and wouldn't work at all for something like the graph of the function y = x3/16.
Suppose we wanted to compute the equation of the line that is tangent to the graph of y = x3/16 at the point (2,1/2). To compute this equation we would need to know the slope of the tangent line at that point.
What makes this difficult is that in order to compute a slope we need two points on the line in question. Here we only have one to work with. One way to estimate the slope is to note that if you tilt the tangent line slightly it will touch the curve at two points instead of just one - if we can figure out where those two points are, we can compute the slope of that secant line and use it to estimate the slope of the tangent line.
As you push the point (a , a3/16 ) ever closer to the point ( 2, 1/2 ), the secant line looks more and more like a tangent line. In the limit, the slope of the secant line becomes
The line passing through (2,1/2) with slope 3/4 has equation
or
Calculus is the study of quantities that change in a continuous fashion. A central question in calculus is how to determine an instantaneous rate of change of some quantity. If that quantity is described via a relation
y = f(x)
the rate of change of y with respect to x over some interval in x is defined to be
Mathematicians use a more compact notation, called the delta notation, to describe changes. In the delta notation the rate of change is
More concretely, we determine this rate of change by picking two nearby x values, x1 and x2, and computing x and y deltas from those two.
To compute an instantaneous rate of change, we compute the rate of change for a very small 
and take the limit as that 
goes to 0.
A slightly more convenient way to do this is to fix one of the two x values and take the limit as the first x moves arbitrarily close to the second x.
Alternatively, you can express x1 as x2 + h for some small number h and take the limit as h goes to 0.
Finally, since we just have one x left, we drop the subscript and consider
we call this quantity the instantaneous rate of change of f(x) at x, or alternatively the derivative of f(x) at x.
A mathematical model for the size of a population of bacteria growing in a culture says that after x hours the size of the population is
What is the instantaneous rate of growth of the population at x = 3 hours?
The limit here can't be solved by algebraic tricks, so we have to estimate it numerically.
| h |  |
|---|---|
| 0.5 | 3.656854249492380 |
| 0.1 | 2.311444133449163 |
| 0.05 | 2.191389441356901 |
| 0.01 | 2.101212570719335 |
| 0.0001 | 2.079657760523101 |
| 0.000001 | 2.079443703673078 |
The rate of change when x = 3 is approximately
The limit
is called the derivative of f(x) at x. We have seen that this limit (or limits very much like it) show up when we try to compute the slopes of tangent lines or estimate instantaneous rates of change. We are going to spend the next several lectures studying this limit and developing techniques for computing it more quickly and easily.
Section 2.7: 1, 2, 5, 7, 8, 15, 16, 22, 27, 28