5128_Ch03_pp098-184
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112 Chapter 3 Derivatives<br />
An Alternative to NDER<br />
Graphing<br />
f(x 0.001) f(x 0.001)<br />
y <br />
0.002<br />
is equivalent to graphing y NDER f x<br />
(useful if NDER is not readily available<br />
on your calculator).<br />
SOLUTION<br />
We saw at the start of this section that x is not differentiable at x 0 since its righthand<br />
and left-hand derivatives at x 0 are not the same. Nonetheless,<br />
0 h 0 h<br />
NDER x, 0 lim<br />
h→0 2h<br />
h h<br />
lim <br />
h→0 2h<br />
0<br />
lim <br />
h→0 2 h<br />
0.<br />
The symmetric difference quotient, which works symmetrically on either side of 0,<br />
never detects the corner! Consequently, most graphing utilities will indicate (wrongly)<br />
that y x is differentiable at x 0, with derivative 0.<br />
Now try Exercise 23.<br />
In light of Example 3, it is worth repeating here that NDER f a actually does approach<br />
f a when f a exists, and in fact approximates it quite well (as in Example 2).<br />
EXPLORATION 2<br />
Looking at the Symmetric Difference Quotient<br />
Analytically<br />
Let f x x 2 and let h 0.01.<br />
1. Find<br />
f 10 h f 10<br />
.<br />
h<br />
How close is it to f 10?<br />
2. Find<br />
f 10 h f 10 h<br />
.<br />
2h<br />
How close is it to f 10?<br />
3. Repeat this comparison for f x x 3 .<br />
.1<br />
.2<br />
.3<br />
.4<br />
.5<br />
.6<br />
.7<br />
X<br />
X = .1<br />
[–2, 4] by [–1, 3]<br />
(a)<br />
Y1<br />
10<br />
5<br />
3.3333<br />
2.5<br />
2<br />
1.6667<br />
1.4286<br />
(b)<br />
Figure 3.17 (a) The graph of NDER<br />
ln x and (b) a table of values. What graph<br />
could this be? (Example 4)<br />
EXAMPLE 4<br />
Graphing a Derivative Using NDER<br />
Let f x ln x. Use NDER to graph y f x. Can you guess what function f x is<br />
by analyzing its graph?<br />
SOLUTION<br />
The graph is shown in Figure 3.17a. The shape of the graph suggests, and the table of<br />
values in Figure 3.17b supports, the conjecture that this is the graph of y 1x. We will<br />
prove in Section 3.9 (using analytic methods) that this is indeed the case.<br />
Now try Exercise 27.<br />
Differentiability Implies Continuity<br />
We began this section with a look at the typical ways that a function could fail to have a<br />
derivative at a point. As one example, we indicated graphically that a discontinuity in the<br />
graph of f would cause one or both of the one-sided derivatives to be nonexistent. It is