Calc02_1day1 - 2.1 Rates of Change Limits Greg Kelly...

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2.1: Rates of Change & Limits Greg Kelly, Hanford High School, Richland, Washington
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Suppose you drive 200 miles, and it takes you 4 hours. Then your average speed is: mi 200 mi 4 hr 50 hr ÷ = distance average speed elapsed time x t = = If you look at your speedometer during this trip, it might read 65 mph. This is your instantaneous speed .
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A rock falls from a high cliff. The position of the rock is given by: 2 16 y t = After 2 seconds: 2 16 2 64 y = = average speed: av 64 ft ft 32 2 sec sec V = = What is the instantaneous speed at 2 seconds?
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instantaneous y V t = for some very small change in t ( 29 ( 29 2 2 16 2 16 2 h h + - = where h = some very small change in t We can use the TI-89 to evaluate this expression for smaller and smaller values of h .
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instantaneous y V t = ( 29 ( 29 2 2 16 2 16 2 h h + - = h y t 1 80 0.1 65.6 .01 64.16 .001 64.016 .0001 64.0016 .00001 64.0002 ( 29 ( 29 { } 16 2 ^ 2 64 1,.1,.01,.001,.0001,.00001 h h h + - ÷ = We can see that the velocity approaches 64 ft/sec as h becomes very small.
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