第5.10节 导数与增量

5.10 DERIVATIVES AND INCREMENTS

In Section 3.3 we found that the derivative of f is given by the limit

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If y = f(x), _______________________,

By definition this means that when the hyperreal number Δx is infinitely close to but not equal to zero, Δy/Δx is infinitely close to dy/dx.

By contrast, the v, δ condition for this limit says intuitively that when the real number Δx is close to but not equal to zero, Δy/ Δx is close to dy/dx.

The ε, δ condition for the derivative can be given a geometric interpretation, shown in Figure 5.10.1. Consider the curve y = f(x), and suppose f ′(c) exists. Draw

The line tangent to the curve at c. For Δx ≠ 0, draw the secant line which intersects the curve at the points (c, f(c)) and (c + Δx, f(c+ Δx)). Then the tangent line will have slope f(c) while the secant line will have slope

___________________________.

The ε, δ condition shows that if we take values of Δx closer and closer to zero, then the slopes of the secant line will get closer and closer to the slope of the tangent line.

EXAMPLE 1 Consider the curve f(x) = x1/3.

Then _________________.

At the point x=8, we have

x=8, f(x) = 2, f ′(x) = _____ = 0.0833…

Thus ______________________________.

This is the slope of the line tangent to the curve at the point (8,2). As Δx approaches zero, the slope of the secant line through the two points (8, 2) and (8+ Δx, (8+ Δx) 1/3) will approach ___ . We make a table showing the slope of the secant line for various values of Δx.

The ε, δ condition for the derivative is of theoretical importance but does not give an error estimate for the limit. When the function f has a continuous second derivative, we can get a useful error estimate in a different way. It is more convenient to work with one-sided limits.

By an error estimate for a limit

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We mean a real function E ( Δx), 0 < Δx ≤ b, such that the approximation g (Δx) is always within E(Δx) of the limit L. In symbols,

|g(Δx) - L| ≤ E (Δx) for 0 <Δx ≤ b.

THEOREM 1

Suppose f has a continuous second derivative and |f ′′(t)|≤ M for all t in the interval

[c, b]. Then:

(i) Whenever c < c + Δx ≤ b, f(c+ Δx) is within ___ MΔx² of f(c) + f ′(c) Δx.

(ii) Whenever c < c + Δx ≤ b, _____________ is within ___ MΔx of f ′(c).

That is, ___ M Δx is an error estimate for the right- sided limit

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There is a similar theorem for the left-sided limit

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With the error estimate ___ M |Δx|.

PROOF Let x= c + x. Then

- M ≤ f ′′(ι) ≤ M for c ≤ t ≤ x.

Integrating from c to t,

_____________________________.

- M (t - c) ≤ f ′(t) - f ′(c)≤ M( t-c ).

Integrating from c to t,

Or