121616949-math.208

# 121616949-math.208 - Figure 9.13 Average velocity Here’s...

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194 Chapter 9 Applications of Integration where the values t i are evenly spaced times between 1 and 3. Once again we are “missing” Δ t , and this time 1 /n is not the correct value. What is Δ t in general? It is the length of a subinterval; in this case we take the interval [1 , 3] and divide it into n subintervals, so each has length (3 - 1) /n = 2 /n = Δ t . Now with the usual “multiply and divide by the same thing” trick we can rewrite the sum: 1 n n - 1 X i =0 16 t 2 i + 5 = 1 3 - 1 n - 1 X i =0 (16 t 2 i + 5) 3 - 1 n = 1 2 n - 1 X i =0 (16 t 2 i + 5) 2 n = 1 2 n - 1 X i =0 (16 t 2 i + 5)Δ t. In the limit this becomes 1 2 Z 3 1 16 t 2 + 5 dt = 1 2 446 3 = 223 3 . Does this seem reasonable? Let’s picture it: in ﬁgure 9.13 is the velocity function together with the horizontal line y = 223 / 3 74 . 3. Certainly the height of the horizontal line looks at least plausible for the average height of the curve. 0 1 2 3 0 25 50 75 100 125 150 ..........................................................................................................................................................
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Unformatted text preview: .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 9.13 Average velocity. Here’s another way to interpret “average” that may make our computation appear even more reasonable. The object of our example goes a certain distance between t = 1 and t = 3. If instead the object were to travel at the average speed over the same time, it would go the same distance. At an average speed of 223 / 3 feet per second for two seconds the object would go 446 / 3 feet. How far does it actually go? We know how to compute...
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