5128_Ch03_pp098-184

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