1/3/2018 Calculus Integrate

Ajune Wanis Ismail

PhD in Computer Graphics | Augmented Reality Technology

Profile Calculus Integrate

Biography
Three examples of a type of problem that arises in various contexts are the
cost function C (x) if marginal cost C'(x) is known; find the population P(t) o
mivielab
the rate P'(r) at which the population is changing is known; find the displac
VicubeLab RG object at time t if the velocity v (t) = s'(r) is known.

MaGICX
Notice that all these problems share the same basic format: to find f(x), give
Former GMM problems are solved by anti-differentiation. An elementary example from b
a manufacturer who determines that over an initial period of production. t
production increases linearly and is given by C ‘(x) = 2x. We shall try to find
function C (x) for which C'(x) = 2x. Although we have no analytical procedu

Classes C(x), it should be clear that the cost function C(x) = x2 will give us the known
2x. But other cost functions will work as well. For example,
Teaching

Resources Material

Master/PhD Supervision

and in fact for any number a,

Research
Thus, any cost function of the form C (x) = x2 + a will give the desired margi
Research Areas/Interest
2x; more information is needed to determine a specific value for a. We shal
moment. The process we are now considering is called antidifferentiation.
Current Projects
can be stated as follows:
Completed Projects

Definition
For a given function f(x), a function g such that

Publication
Journal Papers is called an antiderivative of f. The process of finding such a function g is ca
antidifferentiation. Some mathematicians prefer to call this process indefin
Conference Proceedings
simply integration for reasons that will become apparent in later sections.
Books and Book Chapters

In our introductory example, each of the cost functions x2, x2 + 1, and x2 + 1
Technical Reports and
Other Publications of f(x) = 2x; moreover, C(x) = x2 + a is an antiderivative of f(x) = 2x for any c
whenever g (x) is an antiderivative of f(x), so is g (x) + a for any number a, s

Supervision
It is possible to prove the following even stronger result:

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Undergraduate PSM If g is any antiderivative of f, then every other antiderivative off must have
Projects some number a.
Thus, we can think of g(x) + a as the most general antiderivative of f. Conse
SCSV Admission
general antiderivative of f is not a single function but rather a class of funct
SCSV CGMA depend on a.

Programming Expert CGMA

The German mathematician Gottfried Wilhelm Leibniz (1646-1716) introdu

Useful Links (read as “the antiderivative of f” or “the indefinite integral of f”) to represe
antiderivative of f. Thus, if g is any antiderivative of f, then for any number
UTM

CGMA

Ajune@GMM
In summary:
mivielab

Example 1

Example 2

Example 3

The number a that arises in antidifferentiation is often called an “arbitrary

reasons which will become apparent later, it is also called a “constant of int
examples we have used the letter a to designate this constant, but in practic
(We used the letter a instead of c for our initial illustration involving cost si
denote cost.) The following example gives us an insight into the significance
constant.

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Example 4

Suppose that during the initial stages of production the marginal cost to pro
C ‘(x) = 2x dollars per unit. This time, suppose the manufacturer also knows
production, C(0), is $500. Find the corresponding cost function C (x).

We have already seen that any cost function for this marginal cost must be
x2 + a for some constant a. Since

C (0) = 500 = 02 + a = a,

we have a = 500. Thus, the cost function is given by C(x) = x2 + 500

From this example, we see that the arbitrary constant c is the fixed cost of p
only the marginal cost cannot tell us what that fixed cost is; the fixed cost is
information. Each of the cost functions corresponding to a marginal cost of
the form

C(x) = x2 + (fixed cost).

The following two results are very useful in the evaluation of antiderivative
real number and c is a constant of integration.

Note that Rule (2) holds for n != – 1. and Rule (3) covers the case that n = -1.
use Definition (1) as follows:

To verify Rule (3), recall that

Example 5 Use Rule (2) to evaluate each antiderivative:

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Solutions:

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