# Graphing exponential functions

#### otownsend

Hi,

I hope someone can help.

I'm trying to calculate what the instantaneous rate of change would be at year 2 for the following table:

I'm suppose to plot these points on a graphing calculator and then find the values of y (students per computer) that are close to year 2, then use these points to calculate the instantaneous rate of change. The only problem is that I don't have a graphing calculate on hand that has the power to do this. Are there any only resources that could be a close substitute? Please let me know.

#### skeeter

MHF Helper
Data does not model an exponential function, rather a power function.

Conducted a power regression at this online site ...

Power regression Calculator - High accuracy calculation

A power regression is of the form $y = Ax^B$. You may input the data from the table and conduct it again if you like.

regression results attached ...

Instantaneous rate of change in units of (students per computer)/(year) at year 2 is $\approx -35$

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#### HallsofIvy

MHF Helper
I don't understand what you would expect a calculator to do for you. The x values "closest to 2" are x= 1, x= 2, and x= 3. The "difference quotient" between x= 1 and x= 2 is $$\displaystyle \frac{75- 125}{2- 1}= -\frac{50}{1}= -50$$. The "difference quotient" between x= 2 and x= 3 is $$\displaystyle \frac{50- 75}{3- 2}= -\frac{25}{1}= -25$$. You can combine those and so an get an approximate derivative at 2 by averaging them: $$\displaystyle \frac{-25- 50}{2}= -\frac{75}{2}$$.

#### otownsend

Okay all the responses above make sense. I got the same results as well. I think my textbook is just wrong because it says that the answer is -12.378.

#### otownsend

since the textbook was wrong about instantaneous rate of change (IROC) at year 3, can someone please confirm with me what the IROC was at year 2 and 7?

#### skeeter

MHF Helper
since the textbook was wrong about instantaneous rate of change (IROC) at year 3, can someone please confirm with me what the IROC was at year 2 and 7?
... your original post asked about the rate of change at year 2.

The rate of change for year 3 should be in the neighborhood of -19

The rate of change for year 7 should be in the neighborhood of -2.5

#### otownsend

Thanks for the clarification!