Applied Calculus Math 215 - University of Hawaii

2 CHAPTER 0. A PREVIEW
time by t (say measured in hours) and the size of the population by P (say
measured in millions of bacteria), we denote this function by P (t). You like
to know at what rate the population is changing at some fixed time, say at
time t0 =4.
• For a straight line, the rate of change is its slope.
We like to apply the idea of rate of change or slope also to the function P (t),
although its graph is certainly not a straight line.
What can we do? Let us try to replace the function P (t) by a line L(t),
at least for values of t near t0. The distance between the points (t, P (t))
and (t, L(t)) on the respective graphs is
(1)
E(t) =|P(t)−L(t)|.
This is the error which we make by using L(t) instead of P (t) attimet.We
will require that this error is “small” in a sense which we will precise soon.
If a line L(t) can be found so that the error is small for all t in some open
interval around t0, thenwecallL(t) the tangent line to the graph of P at
t0. The slope of the line L(t) will be called the slope of the graph of P (t) at
the point (t0,P(t0)),ortherateofchangeofP(t)atthetimet=t0.
P(t)
60
55
50
45
40
3.8 3.9 4.1 4.2 t
Figure 2: Zoom in on a point.
P(t)
200
150
100
50
1 2 3 4 5 6 t
Figure 3: Graph & tangent line
Let us make an experiment. Put the graph under a microscope or,
on your graphing calculator, zoom in on the point (4,P(4)) on the graph.
This process works for the given example and most other functions treated
in these notes. You see the zoom picture in Figure 2. Only under close