# determine the average rate of change for the number of bacteria and the rate at which the colony is increasingtime in days 0 1, ...

determine the average rate of change for the number of bacteria and the rate at which the colony is increasing

time in days 0 1, 2, 3, 4, 5, 6,

Population 50, 153, 234, 357, 547, 839, 1280

a) determine the average rate of change for the number of bacteria over the 6 day period.

b) estimate the rate at which the colony is increasing at day 2 .

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### 2 Answers

The average rate of change is

= (Final bacterial count - Initial Count)/(total time)

The average rate of change = (1280-50)/6

= 205 bacteria per day.

To find the rate of increase of colony, the most sitable method is to use the central derivative approximation formula.

The central derivative approximation says,

if `x_(n-1)` ,` x_n` and `x_(n+1)` are subsequent data of a data set and they are equally spaced by an interval of h,

`y'(x_n) = (y(x_(n+1)) - y(x_(n-1)))/(2h)`

This gives,

`r_2 = (357-153)/(2*1)`

`r_2 = 102 ` bacteria per day.

Therefore the rate of increase of colony at day 2 is 102 bacteria per day

first calculate change in bacteria number-153-50=103,234-153=81, 357-234=123,547-357=190,839-547=292,1280-839=441.then add all the numbers and divide it by 6.then answer will be 103+81+123+190+292+441/6=278.5