Unformatted text preview: ine it. 127 In Algebra we are familiar with the concept of discrete summation of a finite
Algebra
discrete
number of terms. For example :
n
1
Σ /2 i where n is finite.
i=0
We are also familiar with the discrete summation of an infinite number of terms.
discrete summation
discr
For example, the geometric series :
∞
1
Σ /2 i
i=0
We could have defined a more general case like :
∞ Σ f(x i ) i=0 Notice that even though the summation involves infinitely many terms, they are
COUNTABLE. Hence we may identify each term using a discrete subscript. Recall
discrete
discr
the difference between the set of natural numbers N that is COUNTABLE, and the
continuous
contiguous set of real numbers R that is UNCOUNTABLE. In the continuous
contin
summation
summation over an interval [a, b] we must include each and every real number or
point or instant in [a, b]. Since there are UNCOUNTABLY many of them it does NOT
make sense to use a subscript. We know only the end points a and b of the interval
[a, b]. So we evolve the notation for the continuous summation in 3 steps :
continuous
n Σ i=0 discrete summation
of FINITELY many terms ⇒ n →∞ Σ i=0 discrete summation
of COUNTABLY many terms ⇒ b ∫
a continuous summation
of UNCOUNTABLY many terms For the continuous summation to be of practical value it must converge to some
continuous
definite finite quantity. This fundamental property, in an intuitive way, is known as
integ
integr able. We give an informal definition of this. 128 25. F(x
ANTIDERIVA
{f(x)
f(x)}
25. F( x ) = ANTIDERIVATIVE {f(x)}
Suppose we know the function that expresses the INSTANTANEOUS RATE OF
CHANGE, are we able to determine the function that expresses the CHANGE ?
Suppose we know that the function y’(t) that describes the ver tical speed i.e.
the INSTANTANEOUS RATE OF CHANGE in height is :
dy(t)
.
= u. si n θ  g t
dt
Are we able to determine the function y(t) = u. si n θ . t  1 g t 2
.
2
y’(t) = that describes the CHANGE in height ? polynomial
We could mechanically say: dy(t) is a polynomial o f the form ax + b
pol
dt
So y(t)...
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 Fall '09
 TAMERDOğAN
 Limit, Δx

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