74
CHAPTER 1
.
.
Limits and Continuity
1-2
1.1
A BRIEF PREVIEW OF CALCULUS: TANGENT LINES
AND THE LENGTH OF A CURVE
In this section, we approach the boundary between precalculus mathematics and the calculus
by investigating several important problems requiring the use of calculus. Recall that the
slope of a straight line is the change in
y
divided by the change in
x
. This fraction is the
same regardless of which two points you use to compute the slope. For example, the points
(0, 1), (1, 4), and (3, 10) all lie on the line
y
=
3
x
+
1. The slope of 3 can be obtained from
any two of the points. For instance,
m
=
4
−
1
1
−
0
=
3o
r
m
=
10
−
1
3
−
0
=
3
.
0.96
0.98
1.00
1.02
1.04
1.90
1.95
2.00
2.05
2.10
y
x
FIGURE 1.3
y
=
x
2
+
1
In the calculus, we generalize this problem to ﬁnd the slope of a
curve
at a point. For
instance, suppose we wanted to ﬁnd the slope of the curve
y
=
x
2
+
1at the point (1, 2). You
might think of picking a second point on the parabola, say (2, 5). The slope of the line through
these two points (called a
secant line;
see Figure 1.2a) is easy enough to compute. We have
m
sec
=
5
−
2
2
−
1
=
3
.
However, using the points (0, 1) and (1, 2), we get a different slope (see Figure 1.2b):
m
sec
=
2
−
1
1
−
0
=
1
.
y
0.5
1
1.5
2
2.5
±
0.5
±
2
2
4
6
x
y
0.5
1
1.5
2
2.5
±
0.5
±
2
2
4
6
x
FIGURE 1.2a
Secant line, slope
=
3
FIGURE 1.2b
Secant line, slope
=
1
For curves other than straight lines, the slopes of secant lines joining different points are
generally
not
the same, as seen in Figures 1.2a and 1.2b.
If you get different slopes using different pairs of points, then what exactly does it mean
for a curve to have a slope at a point? The answer can be visualized by graphically zooming
in on the speciﬁed point. Take the graph of
y
=
x
2
+
1 and zoom in tight on the point
(1, 2). You should get a graph something like the one in Figure 1.3. The graph looks very
much like a straight line. In fact, the more you zoom in, the straighter the curve appears
to be and the less it matters which two points are used to compute a slope. So, here’s the
strategy: pick several points on the parabola, each closer to the point (1, 2) than the previous
one. Compute the slopes of the lines through (1, 2) and each of the points. The closer the
second point gets to (1, 2), the closer the computed slope is to the answer you seek.