Estimate the instantaneous rate of growth (in locations/year) in 2014 by measuring the slope of a tangent. (Round your answer to the nearest integer.)

Question

he number N of locations of a popular coffeehouse chain is given in the table (The numbers of locations as of October 1 are given.)

The sides with the numbers (16,680...) represent N.

Year
2008	16,680
2010	16,858
2012	18,066
2014	21,366
2016	25,085

Estimate the instantaneous rate of growth (in locations/year) in 2014 by measuring the slope of a tangent. (Round your answer to the nearest integer.)

Answer 1

In order to find the instantaneous rate of change during 2014, you must find how much the number of coffeehouses has changed between the time before and the time after 2014, that is, 2012 and 2016, respectively.

2016 coffeehouses - 2012 coffeehouses = 25,085 - 18,066 = 7,019

Thus, 7,019 coffeehouses were added in between 2012 and 2016.

Because 2014 is in between 2012 and 2016, we can estimate the average rate of growth of the number of coffeehouses by dividing the number of coffeehouses added by the number of years past within the given interval.

2016 - 2012 = 4 years

7,019 coffeehouses added between 2012 and 2016 / 4 years past between 2012 and 2016 = 1,754.25 coffeehouses added per year between 2012 and 2016.

Thus, the instantaneous rate of growth in 2014 can be estimated at 1,754.75 coffeehouses per year.