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

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

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

Hi Sophia,

When you're trying to measure the slope of the tangent given a table, you want to measure the slope

(y2-y1)/(x2-x1) of the point you're interested in and the data point before it as well as the point you're interested in and the data point after it. You can then average those two slopes to estimate the instantaneous rate of growth.

In the case of your problem, that date you're interested in is 2014, so you can calculate the slope for 2012 to 2014:

(21366-18066)/(2014-2012) = 1650

You can calculate the slope for 2014 to 2016:

(25085-21366)/(2016-2014) = 1726

The average of these two slopes estimate the instantaneous rate of growth for 2014:

(1650+1726)/2 = 1688 locations per year

Hope this helps!