Average Rate of Change, Derivatives (graduation)

[geekunite]

A high school principal is concerned about the drop in percentage of students who graduate from her school, shown in the following table:

Year Entered School, t
Percent Graduating, P

t | P
1992| 62.4
1995| 54.1
1998| 48.0
2001| 43.5
2004| 41.8

a) Calculate the average rate of change of P for each of the three-year intervals between 1992 and 2004.

b) Does the second derivative appear to be positive or negative between 1992 and 2004?

c) Explain why the values of P and the first derivative are troublesome to the principal.

d) Explain why the sign of the second derivative and the magnitude of the first derivative in the year 2001 may give the principal some cause for optimism.

Thank you so much for helping.

WORK DONE :

a) Average Rate of Change 1995 = -2.4 P/t
Average Rate of Change of 1998 = -1.77 P/t
Average Rate of Change of 2001 = -1.03 P/t

b) t| 1992 | 1995 | 1998 | 2001
f''(P)|-1.38 |-1.02 |-0.75 | -0.28
The second derivate appears to be negative.

c) The values of P and dP/dt are troublesome to the principal because as the years go by, the percentage of students that graduate is decreasing.

d) The rate is slowing down and will likely reverse at some point so the graduation rate will begin to increase.

 

[geekunite]

Could anyone please help me with this problem? I'm really stuck, and not sure if I did any of the parts to the problem correctly. Thank you!!

 

[stapel]

a) I haven't checked your numbers, but the trend is obviously downward, though at a clearly slowing rate, so your values seem reasonable.

b) How did you compute the second derivative? Note: Since a negative second derivative means the original function was concave down, then the values in the table would have to be decreasing by growing amounts. Does this match the tabulated data?

c) Um... Your answer seems only to apply to the tabulated data (that is, to P). The school principal is attempting also to draw conclusions regarding future trends based on underlying trends exposed by the derivatives (that is, dP/dt). What can you say about this?

d) You will need to correct your values for d(dP/dt)/dt before being able to say anything clear about this.

Eliz.