Limit step finally when we let t2 coincide with t1 we

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Unformatted text preview: differentiate each of the various functions just once and build a Table of Derivatives. Deriv tiv We then look at more complicated functions made up from elementary functions. To differentiate these combinations of other functions we have a Set of Rules . Set We also look at the Units of Measur e in differ entiation and integr ation . A differentia integ Units Measure dif entiation inte calculation usually has two parts: an operation and a units of measur e. The opera oper units measure operation with a function or expression as an input operand produces as output function expression another function or expression. When we evaluate this output expression we get a function expression value. The output expression or value has a units of measure associated with expression value units it. For example, we may perform an operation to find the expression π r 2 that tells us the area of a circle. When we evaluate this expression π r 2 , say with r = 2, we get the value 4 π . The expression π r 2 or the value 4π has units of measure units measure [meter2 ] associated with it. We assume we are dealing with WELL-BEHAVED functions : SINGLE VALUED, CONTINUOUS and DIFFERENTIABLE. We have seen the first two properties ( SINGLE VALUED and CONTINUOUS ) and why they are necessary. We conclude this part by looking at the DIFFERENTIABLE property. 56 INSTANT ANTANEOUS RATE 1 2 . INSTANTANEOUS RATE OF CHANGE of h e i g h t y(t2) y(t1) h1 (0 , 0) y(t) y (t) t1 h2 t2 T time If we measure the height at any chosen instant, say t 1 , then in terms of the height function y(t) = u . s i n θ . t -- 1 g t 2 t he AVERAGE SPEED over -2 time t = 0 to t 1 is : y (t ) distance = u . s i n θ -- 1 g t -= 2 time t We may ask: what is the actual ver tical speed at the time instant t 1 ? i nstant We shall do this in three steps. VERAGE step: 1. AVERAGE step: If we take a par ticular small time interval, say t 1 t o t 2 , we could be more precise. AVERAGE VERTICAL SPEED from time t1 to t2 is : CHANGE in height C HANGE in time ∆ y y(t ) − y(t ) = ∆ t = 2t − t 1 2 1 57 step: 2. TENDS TO s...
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