Just in time for the holidays, a brand new toy has hit the market, the My First Cliff Diving Kit. It is all the rage, because it comes with everything you need to be an effective cliff diver: swim trunks, neck brace, legal documents for naming next of kin, and very detailed one-paragraph instruction sheet about how to cliff dive safely. Below, you'll find a chart representing the sales figures for the each month it has been available. Use that chart to answer the questions that follow.

(a) Assuming that you can draw a continuous graph connecting all the data points, write the equation of line L, the tangent line to the sales graph when x = May 2002.

(b) What is the average rate of sales between November 2001 and September 2002?

(c) Draw a rough sketch of the graph representing the rate of sales between Oct 2001 and Oct 2002.

Difficulty:

Solution

(a) To write the equation of a line, we need two things: a point and a slope. We'll need to assign some points, since the independent variables are months and not numbers. I will assign October 2001 an x-value of 0 and October 2002 an x-value of 12. This gives us a graph like below:

Therefore, the point representing May 2002 is the coordinate (7,1559). We'll have to estimate the tangent there. I'll do that by calculating the slope between that point and the point immediately preceding it, (6,1509):

Therefore, a good approximation for the tangent line would be , according to point-slope form. You could also approximate the derivative by using the point immediately after May 2002, and you'll get a slope of 128 rather than 50. That seems like a huge difference, but since we don't have any intermediate points, who's to know which is the better approximation?

(b) The average rate of change is calculated using the slope of the secant line connecting the points: (1,634) and (11,2445):

Therefore, the kits are selling better overall on the entire time period when compared with how they were selling in May 2001, no matter what derivative approximation you used.

(c) You can use the same process we did in (a) to approximate each derivative, and you'll get something that looks roughly like this:

Notice that the graph dips below the x-axis on roughly the interval (2,4) because the data seems to be decreasing there. the faster the data points increase, the dreater the height of the blue derivative graph.