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 ? O
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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
Quoted Instructions for "Calculus and Definitions" Assignment:
Introduction:
As you begin to study more advanced calculus topics, it is important to understand concepts that include the use of indefinite integrals, definite integrals, and sequence analysis methods within a variety of applications. The purpose of this task will be to help you review these concepts within selected applications.
Task:
Write a brief essay (suggested length of 4*****5 pages) in which you do the following:
A. Define the concept of the indefinite integral using its formal definition.
1. Explain, in language that high school students could understand, what the formal definition of an indefinite integral is.
2. Provide an example that is relevant to the discussion and supports all claims.
B. Define the concept of the definite integral using its formal definition.
1. Explain, in language that high school students could understand, what the formal definition of a definite integral is.
2. Provide an example that is relevant to the discussion and supports all claims.
C. If you use sources, include all in-text citations and references in APA format
How to Reference "Calculus and Definitions" Assessment in a Bibliography
“Calculus and Definitions.” A1-TermPaper.com, 2012, https://www.a1-termpaper.com/topics/essay/advanced-calculus/7575837. Accessed 27 Sep 2024.
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