Average Rate of Change

[besweeet]

I need some help with this problem:
A company introduces a new product for which the number of units sold S is given by the equation below, where t is the time in months.

I need to:
a) Find the average rate of change of S(t) during the first year (rounded to 1 decimal place).

And:
b) During what month does S'(t) equal the average rate of change during the first year?

I'm not too sure what to do for the first one. Some people have told me to find the derivative of the problem then plug 12 in for t, but that's not right. I could probably find B after I find A.

 

[wjm11]

A company introduces a new product for which the number of units sold S is given by the equation below, where t is the time in months.


I need to:
a) Find the average rate of change of S(t) during the first year (rounded to 1 decimal place).

And:
b) During what month does S'(t) equal the average rate of change during the first year?

I'm not too sure what to do for the first one. Some people have told me to find the derivative of the problem then plug 12 in for t, but that's not right. I could probably find B after I find A.

The average rate is just the slope of a line between two points, in this case the points are at t = 0 and t = 12. Find the function value at those two times, then calculate the slope.

For part b, use the derivative. Set the derivative equal to the slope found in part a) and solve for t.

 

[besweeet]

Thanks for the reply.

I got 189.4736 for s(12), then 56.25 for s(0). I subtracted S(12) and S(0), and got 133.223684211. I then divided that by 12 and got 11.1, which is incorrect (for part A).

 

[wjm11]

I got 189.4736 for s(12), then 56.25 for s(0). I subtracted S(12) and S(0), and got 133.223684211. I then divided that by 12

Your approach is correct, though your calculations are not. Your s(12) value is wrong; it should be 182.8125. Please be careful and double-check your work.

BTW, this problem is a demonstration of the Mean Value Theorem.

 

[besweeet]

Ok, so I got 182.8125 for S(12) and 56.25 for S(0). Subtracting the two and dividing by 12 gives me 10.6 (rounding to one decimal).

So I took the derivative of the original problem and set it equal to 10.6. I got two answers; -11.9799 and 3.97993.

This is the last submission I have for this problem, so I need to make sure this is right ahead of time. Is part A correct? For part B, I just have to choose the month from a drop-down menu.

 

[wjm11]

Ok, so I got 182.8125 for S(12) and 56.25 for S(0). Subtracting the two and dividing by 12 gives me 10.6 (rounding to one decimal).

So I took the derivative of the original problem and set it equal to 10.6. I got two answers; -11.9799 and 3.97993.

This is the last submission I have for this problem, so I need to make sure this is right ahead of time. Is part A correct? For part B, I just have to choose the month from a drop-down menu.

Yes, t = 4 is correct. Note that t = -12 is outside the domain you are considering. Do you understand the connection to the Mean Value Theorem?

 

[besweeet]

Yes (quite simple... Just needed a little guidance).

The correct month is actually May. I don't see how that could be...

 

[wjm11]

The correct month is actually May. I don't see how that could be...

t = 4 means at the completion of 4 months, so that would mean the end of April/start of May, I guess.