Average Rate of Change of a Function
Let's say we have a profit function, P(x) = 2x3 - 20x2 -18x - 5000 where x represents the number
of items sold. Say we were interested how the profit changes, on average, when the number of items sold are between 50 & 60.
Translation: Find the average rate of change of P(x) between x=50 & x=60.
Solution: The average rate of change of any function f(x) between two values of x, say x=a and x=b is given by the slope of the line (secant line) connecting
these points. Using the functional notation & computing the slope by formula we get: f(b) - f(a) divided by b - a. This is just the y-change divided by the x change between the two points
(a,f(a)) and (b,f(b)) on this curve.
In this problem, we are using the P(x) function and the values of a=50, b=60. So, your two points are (50,P(50)) and (60,P(60)).
You will get P(60)=353920 and P(50)=194100 by direct substitution into the equation. But, the procedure below is much easier.
To make life easy, code in this equation in your calculator by pressing the Y= button. Powers on x can be coded in by pressing the ^ button then the power. Since you have your table already set ( i.e., ask for the Independent
variable & auto for the dependent variable) you don't have to bother with that again. Then press 2nd, graph to display your table. Insert x=50, enter. P(50) will appear in the Y column. Likewise, for x=60. Entering any
x value will display the y-value (functional value) in the 2nd column.
Now, simply compute P(60) - P(50) divided by 60 - 50. The 2nd difference of 10 can be done in your head. Be careful, the difference in the bottom must be in the correct order (i.e., Since P(60) is first on top, 60 must be first on the bottom). If you do
everything correctly, you should get the answer of $15,982 per item (on average). These items contain diamonds.
Extra: fool around with the tables values (input different x's & see if you can find the minimum number of items to sell for a profit. That will be 19.
You could also press Graph to get a sketch of this curve. If you fool around with the x window (i.e., take xmin at 0 & max at 25 and you'll be able to see where this curve crosses the x-axis). This is the break even point. Using trace, you'll be able to estimate the x value
by moving the trace point along the curve. But, fooling with the table & looking at the 2nd column will give you where the curve is as well. In that case, when the number in the 2nd column (profit) changes from negative to positive gives the answer for x.