alittlebitofcalculus-pdf-january2011-111112001007-phpapp01

Physicists and chemists work with molecules

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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 point instant have the special word inf initesimal to describe it. Understanding the behavior infinitesimal i nf of a function over an inf initesimal interval will help us to: infinitesimal : i nf 1. Formulate an expression to describe its behavior at a point or instant. point 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 h e i g h t y(t2) y(T ) = maximum / 2 y(t3) y(t1) h1 y(t4) h3 h2 h4 (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. Example: 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 ? xpression 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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