Mean value theorem
The derivative of a function f at a point ξ$$f'\left( \xi \right) = \mathop {\lim }\limits_{\Delta x \to 0} {\rm{ }}{{f\left( {\xi + \Delta x} \right) - f\left( \xi \right)} \over {\Delta x}},$$ is the slope of the line tangent to the graph of f at the po
- PDF / 5,969,027 Bytes
- 55 Pages / 439.32 x 666.12 pts Page_size
- 4 Downloads / 222 Views
Mean value theorem
The derivative of a function
f
at a point ~
f'(~) = lim f(~ ~x---+o
+ ~x) -
f(O ,
~x
is the slope of the line tangent to the graph of f at the point P = (~, f(~)). Restricting to ~x > 0 we see that f' (0 is the limit of the slopes of secants PQ, as Q -+ P from the right (Fig. 7.1 a). Similarly, for ~x < 0 we get f' (~) as the limit of slopes of secants RP, as R -+ P from the left (Fig. 7.1 b). These are local results, for points which approach P in the limit. A related question of a global nature is: Let A = (a, f(a)) and B = (b, feb)) be any two points on the graph of a differentiable function f. Is there a point ~, a < ~ < b, where the derivative f' (~) equals the slope of the secant AB? See Fig. 7.1 c. The answer "yes" is known as the mean value theorem. It is one of the most important results in calculus (simple questions may have profound answers). In this chapter we study the mean value theorem, and some of its consequences and applications. The applications selected here include the rule of l'Hospital (in Sect. 7.2), Taylor's theorem (Sect. 7.3) and antiderivatives (Sect. 7.4). The mean value theorem is also used in the study of iterative methods (Sect. 7.5), in particular Newton's method (Sect. 7.6), and fixed points (Sect. 7.7). Y
a
b
tangent
c
Fig. 7.1 a-c. Tangents and secants. a The derivative at P is the limit of the slopes of the secants PQ as Q ---+ P. b The derivative at P is the limit of the slopes of the secants RP as R ---+ P. c A secant AB and an "intermediate" point P where the derivative equals the slope of AB A. Ben-Israel et al., Computer-Supported Calculus © Springer-Verlag/Wien 2002
225
7.1 Mean value theorem
7.1 Mean value theorem The mean value theorem has several versions, the easiest to state and prove is the following result due to Rolle l . Theorem 7.1 (Rolle's theorem). If f is continuous in the closed interval [a, b], differentiable in the open interval (a, b), and f(a) = feb), then there is at least one; E (a, b) where !'(;) = o.
Proof If f is constant, then the result is obviously true. Otherwise there are points x E (a, b) where either f(x) > f(a) or f(x) < f(a). In the first (second) case, the function f attains its maximum (minimum) in [a, b] (see Theorem 3.52 at an interior point ;). The necessary condition of Theorem 6.7 then guarantees that !'(O = o. D The geometrical interpretation of Rolle's theorem is: Given a closed interval, and a differentiable function f with equal values at the endpoints, there is (at least) one interior point of the interval where the graph of f has a horizontal tangent (Fig.7.2a). Similarly, between any two points A and B on the graph of a differentiable function, there is at least one point of the graph where the slope of the tangent line equals the slope ofthe lineAB (see the points;l and;2 in Fig. 7.2b). This result, due to Lagrange, is stated formally as follows. Theorem 7.2 (Mean value theorem). If f is continuous in the closed interval [a, b] and differentiable in the open interval (a, b), then there is a
Data Loading...
