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Interpolation and Approximation Theory
Published in A. C. Faul, A Concise Introduction to Numerical Analysis, 2018
For t = xj the first term vanishes, since f(xj) = fj = p(xj), and by construction the product in the second term vanishes. We also have ϕ(x) = 0, since then the two terms cancel. Hence ϕ has at least n + 2 distinct zeros in [a, b]. By Rolle’s theorem if a function with continuous derivative vanishes at two distinct points, then its derivative vanishes at an intermediate point. Since ϕ ∈ Cn+1[a, b], we can deduce that ϕ′ vanishes at n + 1 distinct points in [a, b]. Applying Rolle again, we see that ϕ″ vanishes at n distinct points in [a, b]. By induction, ϕ(n+1) vanishes once, say at ξ ∈ [a, b]. Since p is an nth degree polynomial, we have p(n+1) ≡ 0. On the other hand, dn+1dtn+1∏i=0n(t−xi)=(n+1)!.
Differentiation of functions of one or more real variables
Published in Alan Jeffrey, Mathematics, 2004
If f is not differentiable at even one interior point of (a, b) then Rolle’s theorem cannot be applied. Our counter-example in this instance is the simple function f(x) = |x| with − 1 ≤ x ≤ 1. This function is everywhere continuous, and is differentiable at all points other than at the origin, but there is certainly no point x = ξ on [−1, 1] at which f′ = 0. The graph of this function is shown in Fig. 5.9, with one of a function g(x) not satisfying the conditions of the theorem but for which the result happens to be true.
Series expansions and their uses
Published in C.W. Evans, Engineering Mathematics, 2019
Now g(a) = 0 and g(b) = 0, and so by Rolle’s theorem there exists c ∈ (a, b) such that g′(c) = 0. That is, f′(c)=f(b)−f(a)b−a
Pre-service mathematics teachers using Geogebra to learn about instantaneous rate of change
Published in International Journal of Mathematical Education in Science and Technology, 2023
Vilmar Gomes da Fonseca, Ana Cláudia Correia Batalha Henriques
This concept is also essential for the proof of diverse calculus theorems associated with the derivative concept, such as those involving differentiable functions in a closed interval . For example, the Lagrange mean value theorem which guarantees the existence of a point in the graph of a function whose derivative corresponds to the slope of the secant line to this graph that passes through the points and , that is, such that ; and the Rolle theorem to ensure that a function , with , has at least one maximum or minimum point, that is, such that (Abramovitz et al., 2009). The proof of these theorems is important in promoting conceptual understanding of the IRC, as well as providing an opportunity for students to develop the ability to think abstractly about the concept of limit (Kidron & Tall, 2015). In fact, the learning of the IRC involves, among other things, recognizing the existence of the limit at the point and the differential as assumptions of its applicability; the validation of these assumptions for the limit’s algebraic calculation, which may include the resolution of indetermination type of limit; and the application of this knowledge to solve problems (Ocal, 2017; Sahin et al., 2015).