A small company estimates that when it spends x thousand dollars for advertising in a year, its annual sales will be described by s(x) = 60 − 40e−0.05x thousand dollars. The four most recent annual advertising totals are give in the following table. Year 1 2 3 Dollars 14,500 16,000 18,000 i)Estimate the current ii)the current rate of change of sales.
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I read s(x)=60-40e^(-0.05x), where x is advertising cost. I read that s is sales in thousands of dollars. Let's look at the table. When x=14.5, 1000s=60-40e^-0.725=$40627; 1000s(16)=$42027; 1000s(18)=$43737. In each case the result shown is amount in dollars, rather than thousands of dollars, hence 1000s.

The rate of change is given by ds/dx=40*0.05e^(-0.05x)=2e^(-0.05x). This is the rate of change of sales revenue with increasing advertising cost. The rate of change per year is ds/dt where t is time in years. ds/dt=2e^(-0.05x)dx/dt. In other words the rate of change in sales revenue per year is related to the rate of change of advertising costs per year. The table has two changes: first is change between years 1 and 2 (1.5); second is change between years 2 and 3 (2). The corresponding change in sales is $1400 and $1710. The average change in advertising costs is 1.75 thousands of dollars; the average change in sales is $1555.

The derivative for each of the three years is $968, $899 and $813, giving an average of $893.

 

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