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# Derivatives, Tangents and Interpreting the Slope of a Regression

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1. The table below presents the net sales (Revenue), R(t) in billions of dollars for Wal-Mart for the period 1994 to 2004 (Wal-Mart's website). Let t = 0 represent 1990.

t 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004
P(t) 63 78 89 100 112 131 156 181 204 230 256

a. Use your graphing utility to find the regression line of best fit for the data. (Hint: Notice here that we said a line here, as in a straight line, not a curve. If you select some kind of curve, for example a parabola, you are mistaken, and you will be unable to correctly the answer the remaining questions in problem 1.)

y = R(t) =

b. What is the slope of the line from part a? Interpret its meaning in the context of the problem. Include units!

m =

meaning:

c. According to your linear model, when will Wal-Mart be making 300 billion dollars in net sales? Show work.

d. What is the y-intercept in the equation in part (a)? Interpret its meaning in the context of the problem. Include units!

y-intercept:

meaning:

e. What is the derivative of the linear equation in part (a)? _______________.

We observe that the derivative of the linear equation is the same as the s_____________ of the line.

2. The profit (in millions of dollars) of a company is modeled by the function . The variable x represents the number of years since 1990.

a. Find P(2) and explain in a sentence what the value P(2) means in this context.

b. Sketch the secant line on the above graph to show the average rate of change in profit from 1992 to 1994

c. What is the average rate of change of the company's profit during the period 1992 to 1994? (hint: what is ?)

d. Draw a tangent line on the above graph to represent the instantaneous rate of change of the profit in 1993. Show your tangent line on the graph.

e. Use the derivative to calculate the slope of the tangent line (the instantaneous rate of change of the profit) of in 1993.

f. Use the derivative to find the value of x that gives a maximum profit between the years 1990 and 2000. Just a number without justifying calculations will result in zero credit on this part, since maximums can be found on calculators without much thinking. You need to show the process of what to do with the derivative to find a maximum to get credit here.

g. In what year (after 1990) did the company start losing money?
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