5128_Ch03_pp098-184

Section 3.2 Differentiability 111
y
m 1 =
f(a + h) – f(a – h)
2h
m 2 =
f(a + h) – f(a)
h
tangent line
Derivatives on a Calculator
Many graphing utilities can approximate derivatives numerically with good accuracy at
most points of their domains.
For small values of h, the difference quotient
f a h f a
h
a – h a a + h
Figure 3.16 The symmetric difference
quotient (slope m 1 ) usually gives a better
approximation of the derivative for a given
value of h than does the regular difference
quotient (slope m 2 ), which is why the
symmetric difference quotient is used in
the numerical derivative.
x
is often a good numerical approximation of f a. However, as suggested by Figure 3.16,
the same value of h will usually yield a better approximation if we use the symmetric
difference quotient
f a h f a h
2h ,
which is what our graphing calculator uses to calculate NDER f a, the numerical
derivative of f at a point a. The numerical derivative of f as a function is denoted by
NDER f x. Sometimes we will use NDER f x, a for NDER f a when we want to
emphasize both the function and the point.
Although the symmetric difference quotient is not the quotient used in the definition of
f a, it can be proven that
lim
h→0
f a h f a h
2h
equals f a wherever f a exists.
You might think that an extremely small value of h would be required to give an accurate
approximation of f a, but in most cases h 0.001 is more than adequate. In fact,
your calculator probably assumes such a value for h unless you choose to specify otherwise
(consult your Owner’s Manual). The numerical derivatives we compute in this book
will use h 0.001; that is,
f a 0.001 f a 0.001
NDER f a
0.002 .
EXAMPLE 2
Computing a Numerical Derivative
Compute NDER x 3 ,2, the numerical derivative of x 3 at x 2.
SOLUTION
Using h 0.001,
NDER x 3 ,2
2.001 3 1.999 3
0.002
12.000001.
Now try Exercise 17.
In Example 1 of Section 3.1, we found the derivative of x 3 to be 3x 2 , whose value
at x 2 is 32 2 12. The numerical derivative is accurate to 5 decimal places. Not bad
for the push of a button.
Example 2 gives dramatic evidence that NDER is very accurate when h 0.001. Such
accuracy is usually the case, although it is also possible for NDER to produce some surprisingly
inaccurate results, as in Example 3.
EXAMPLE 3
Fooling the Symmetric Difference Quotient
Compute NDER x, 0, the numerical derivative of x at x 0.
continued