Unformatted text preview: an element. Very
often scientists work with entities smaller than cells or atoms. However, no matter
how small it is, it is always something tangible.
While molecules, atoms and particles are DISCRETE objects, in Calculus we work with
CONTINUOUS entities like the functions that describe the motion of an object - its
position, speed, acceleration and so on. We call these CONTINUOUS operands.
Mathematicians like to study the behavior of a function at a point or instant and
p oint instant
over a very small interval around a point or instant. The interval is so small that we
have the special word inf initesimal to describe it. Understanding the behavior
of a function over an inf initesimal interval will help us to:
1. Formulate an expression to describe its behavior at a point or instant.
2. Use this expression to extract more information about the behavior of the
function, i.e. is the function increasing or decreasing, at a maximum or
minimum, or changing direction. Let us see an example of a CONTINUOUS operand and the kind of operations we
would like to perform. 1 M OT I VAT I O N
t y(t2) y(T ) = maximum
h1 y(t4) h3
(0 , 0) t 0 t 1 t2 T
2 t3 t4 tn = T time Throw a ball up into the air. Suppose we know the height at any instant in
time, i.e. we know the function y(t) that describes the height. From this we
can estimate the CHANGE in height with respect to time.
CHANGE in height from time t 1 t o t 2 i s simply : h 2 -- h 1
which is : y(t 2 ) -- y(t 1 )
From the function y(t) can we find out the RATE OF CHANGE in height at any
chosen INSTANT ?
What is the ver tical speed or RATE OF CHANGE in height at time instant t1 ?
What is the expression of the INSTANTANEOUS RATE OF CHANGE in height ?
CHANGE in height
We know that ver tical speed =
CHANGE in time
y(t 2 ) -- y(t 1 )
Ver tical speed between t 1 a nd t 2 =
t 2 -- t 1
2 This is the AVERAGE VERTICAL SPEED over the time interval t1 to t2.
No matter how close we take t2 to t1, i.e. no matter how small the time
interval is, we still get only an AVERAGE VE...
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- Fall '09
- Limit, Δx