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A Review of Calculus
Published in Richard L. Shell, Ernest L. Hall, Handbook of Industrial Automation, 2000
The mean value theorem states that if f is differentiable on (a, b) and continuous on [a, b] then there exists some point c inside (a, b) with the property that f(b) −f(a) =f′(c)(b − a). Using this theorem it is not difficult to show that, in fact, Thedefiniteintegralofffromatob=F(b)−F(a)
Differentiation of functions of one or more real variables
Published in Alan Jeffrey, Mathematics, 2004
Our most important application of Rolle’s theorem will be in the derivation of the mean value theorem for derivatives. It is difficult to indicate just how valuable and powerful this deceptively simple theorem really is as an analytical tool. However, something of its utility will, perhaps, be appreciated after studying the reminder of this chapter. We offer only a geometrical proof of the theorem.
Online integral calculators and the concept of indefinite integral
Published in International Journal of Mathematical Education in Science and Technology, 2023
The function satisfies the condition of the Mean value theorem, i.e. it is continuous on and has the derivative on . Therefore, for some point . Since is continuous at the point , the left limit of the difference quotient exists and equals . Similarly, the right limit of the difference quotient exists and equals .
Experimental validation of a compression flow model of Non-Newtonian adhesives
Published in The Journal of Adhesion, 2022
Marvin Kaufmann, Florian Flaig, Michael Müller, Lukas Stahl, Jannis Finke, Till Vallée, Holger Fricke
So far, this expression is independent of the constitutive law. For a Newtonian fluid, that is, a constant viscosity, the partial differential equations are linear with respect to the velocity gradients and can simply be integrated with respect to in order to obtain the velocity profiles. Subsequently, the mean velocities can be computed by means of the mean value theorem. In contrast, for non-linear materials, such as the Yasuda model, [23] the partial differential equations become non-linear with respect to the velocity gradients into -direction and cannot be solved analytically, in general. The Power-Law is an exception, since can be found an analytical solution as shown in Refs. [20,24] At this point, the key idea is to find a discrete reconstruction of the velocity gradients into -direction which can, in turn, be integrated numerically by means of the trapezoidal rule in order to obtain the flow profiles and mean velocities. The discrete reconstruction of the velocity gradients is obtained by solving the non-linear system in Eq. (7) for a finite number of discrete positions into -direction using a Newton-Raphson method. In this way, both shear rate and velocity profiles as well as the mean velocities can be approximated for any kind of Generalized Newtonian fluid.
Some variants of Cauchy's mean value theorem
Published in International Journal of Mathematical Education in Science and Technology, 2020
The first contact of our mathematics or engineering undergraduate students with the mean value theory is in a Differential and Integral Calculus course or in a first Real Analysis course. The first mean value theorem is the famous Lagrange's Mean Value Theorem, which relates the average rate of change of a function at the end of an interval with the value of the derivative of the function at a point in the same interval. The second mean value theorem is Cauchy's Mean Value Theorem, which is a generalization of the Lagrange's Mean Value Theorem, it establishes the relationship between the derivatives of two functions and the variation of these functions on a finite interval.