Differentiation andRiemann Integration - Assignment Example

Exercise 4.4 what is the connection between the calculations in the box above and the area under the graph of x2 between x = 0 and x = 1? And what has that to do with an integral?

If we have a function where f(x) ≥ 0 over an interval a≤ x ≤b, we examine the area of this region that is said to be included in the graph of f(x) and over the interval [a, b] found on the x-axis. The area found under the region marked x = 0 and x = 1 is called the “area found under a curve”. This is an aspect that makes it related to an integral. An integral refers to the associated notion of the ant derivative, a function X whose derivative is x, which is the given function. The use of integrals is a vital part of calculus and was well explained that integral includes rectangles having infinite sums and with infinitesimal width. Riemann stated that this integral basis itself on limiting procedures that appropriate a curvilinear region by approximating its area as it breaks them into thin vertical blocks (Rana, 2002).

Exercise 4.7 Consider the Fundamental Theorem of Calculus (Theorem 5 in\Notes on Riemann Integration").

1. Let x ­[0;1] be the characteristic function of the unit interval. What does the Theorem tell us about the function h where h(x) =Zx1 [0; 1] (t) dt

As explained by the Riemann integration, it is evident that x is a set bound by finite point sets. According to the Riemann integration theorem, the function h is defined as the indicator function, which is equal to the figures that are on opposite sides. This integration explains that you can use continuous functions to find and substitute the figures inside. The characteristics function definition has a formula, which allows us to compute the value of h if we realize the distribution function Z (Taylor, 2006).

Exercise 4.13 you have studied over the last few years how calculus is made rigorous with definitions (of continuity, derivative, integral, convergence, etc) and theorems. Is this necessary and/or important? Why, or why not?

The use of differentiation is an aspect that is fundamental in calculus. This is based on the functions used as they are continuous. These formulas help in the derivation of the rules of the Leibniz integral. The use of functions, numbers, limits, and integration is a vital aspect also in the calculations of calculus (Pfeffer, 2001).