Assessment on "Calculus and Definitions"

Assessment 5 pages (1309 words) Sources: 1+

[EXCERPT] . . . .

In simple terms the Riemann sum is used to define the definite integral of a function. We begin by considering a simple case, whereby the definition of the Riemann integral of a continuous function f over a rectangle R. In this case rather than having a one-variable case, we can overcome the tendency of connecting integration too strongly with anti-differentiation. (Buck 2003)

Formal definition

It is the definition of a function by use of graphs to define the limits; it uses Greek letters epsilon (?) and Delta (?). Epsilon always represents any distance on the limiting side and delta represents the distance on the x- axis. The limits of a given function clearly explain how that given function behaves when it nears the x value.

Consider the following functions g (x) and f (x), this functions are as a result of definition of real numbers. The following relationship exist x ?

This relationship exist only when there is a positive constant C. such that for all sufficiently large values of x, f (x) is at most C. multiplied by g (x) in absolute value. That is, f (x) = O (g (x)) if and only if there exists a positive real number C. And a real number x0 such that

In general the growth rate are of much interest in that the variable x which goes to infinity is often left unstated, and one writes more simply that f (x) = O (g (x)).

Additional explanation indicates that it doesn't matter how close a function can be to a limit, it is always necessary to find the corresponding x value which is closer to the given value and using the new notations of epsilon (?) and delta (?), we make f (x) within ? Of L, the limit, and later determine x within ? Of C. (Bradley et al.,.2000)

Again, since this is tricky, let's resume our example from before: f (x) =x2at x=2. To start, let's say we want f (x) to be within .01 of the limit. We know by now that the limit should be 4, so we say: for ?= 0.1, there is some ? so that as long as, then

To show this, we can pick any delta (?) that is bigger than 0, so long as it works. For example, you might pick .000000001, because you are absolutely sure that if x is within .00000000000001 of 2, then f (x) will be within .01 of 4. This works for. But we can't just pick a specific value for, like .01, because we said in our definition "for every." This means that we need to be able to show an infinite number of s, one for each.

In summary indefinite integration exists when the limits of integration are not given, this means that the upper and the lower limits are not given. Definite integration occurs when the limits are given and therefore you need to calculate the area that is when x=c to x=d

Examples of indefinite integration

You can be given a function and told to verify for example;

Verify that F (x)=is an anti-derivative of f (x) = x2+5

To solve this, first of all we know that F (x) is an anti-derivative of f (x) if and only if F'(x) = f (x)

When F. is differentiated we find that F'(x) = ) +5

=x2+5 = f (x) as required

Examples of definite integration

Let R. be the region under the graph f (x) = 2 xs+1 over the interval 1? x ? 3, compute the area of R. As the limit of the sum.

We can solve this by first subdividing the area under the curve into six approximate rectangles each of width ?x = = . The left end points in the partition 1? x ? 3 are x1= 1, x2= 1+ and similarly, x3 =

Thus the corresponding area of this… READ MORE

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