Advertising (in thousands)	Profit (in thousands)
5	150
7	200
9	250
11	325
13	400
15	450
17	500
19	525

1) Find a cubic model, , for the profit (in thousands), where is the advertising expenditure, in thousands of dollars. Do not alter the inputs. Round each coefficient to 2 decimal places. State the function.

2) Find the rate of change function , including proper units.

3) Using the model, find the instantaneous rate of change when $10,000 is being invested in advertising each month.

4) Using the table data, find the average rate of change when the advertising expenditure increases from $9,000 to $11,000.

5) Using the graph of the model , estimate the -values where the tangent line would be horizontal. Paste a copy of the graph here using these instructions:

a. Graph and set the viewing window to a good fit.

b. Click the icon at the top right that is a box with an arrow in it.

c. Click the button to Export Image.

d. Click Download PNG.

e. Insert the PNG below and adjust the size as needed.

6) Using the graph of the derivative , locate the -values where the derivative is equal to 0 (crosses the -axis). In Desmos, label this graph instead of to avoid confusing Desmos. Paste a copy of this graph here using the above instructions.

7) Are your answers to #5 and #6 similar? Explain why they should be.

8) Using the graph of the model , estimate the -value of the inflection point.

9) Using the graph of the model , estimate the slope of the graph / slope of a tangent line at the inflection point. To do this, follow these steps:

a. Zoom in on the inflection point until the graph looks like a straight line.

b. Estimate two points at / near the inflection point. State the two points here.

c. Compute the slope between the two points. State the slope.

10) Using the graph of the derivative , estimate the -value of the maximum / highest point.

11) Are your answers to #9 and #10 similar? Explain why they should be.