Estimate the instantaneous rate of change at the point indicated.
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Recall that the instantaneous rate of change at x = x0 is the limit of the average rates of change.
For x = x0, the average rate of change of y = f(x) over [x0, x1] has the following formula.
Average rate of change =
where and /gyv4StTnixZTcnynbgrMi0adcL37d1XJwOewtY/zU4we9prEKHQkff9E5b6rMXcANV3O0wRGR7OyJpSvzSK6HjPIihasfwJzK6eSV9Xq0+3P7+1tLvcieviMG5MiIgW.
To estimate the instantaneous rate of change of the given problem, we calculate the average rate of change over smaller and smaller intervals to the nO/+DAdRVq+2oOvmzNgeN0ocEUKz3yEuOTAqpssC58Q= of x = nc1ItEz0kR4=. That is, we find the limit of the average rates of change as x approaches $a.
First calculate the average rate of change over four intervals to the right of x = $a. (Round your answers to six decimal places.)
Interval | [$a,$ap01] | [$a,$ap001] | [$a,$ap0001] | [$a,$ap00001] |
---|---|---|---|---|
Average rate of change | RhjwTClEuoy99C6T | CWO/+PXAHUZw9M4bw+br6Q== | BW3epklUTIVOoCgS43LxhQ== | F8CIPolyEJQLPTb6wPCEFw== |
This table suggests the limit of the average rates of change as x approaches $a from the right is approximately AyAQildZEmgrIPWj (rounded to four decimal places).
Now calculate the average rate of change over four intervals to the left of x = $a. (Round your answers to six decimal places.)
Interval | [$am01,$a] | [$am001,$a] | [$am0001,$a] | [$am00001,$a] |
---|---|---|---|---|
Average rate of change | sAF6ALj0Z4+paJAF | OSbqcGxs4qnq/dTMRDVB2g== | PgSsY9hDTloSRysei0Vhtg== | R78Nu8s7sf9aKF8Uf2d7mQ== |
This table suggests the limit of the average rates of change as x approaches $a from the left is approximately AyAQildZEmgrIPWj (rounded to four decimal places).
Thus, from steps 2 and 3, the limits of the average rates of change from the left and from the right as x approaches $a is $inst.
Thus, the estimate of the instantaneous rate of change is AyAQildZEmgrIPWj (rounded to four decimal places).