Calculus (Derivatives)

Question

11. The length L of the day in minutes (sunrise to sunset) x kilometers north of the equator on June 21 is given by

L = f(x). What are the units of

(a) f'(3000)? (b) f''(3000)?

12. A high school principal is concerned about the drop in the percentage of students who graduate from her

school, show in the following table.

Year entered school, t: 1992 1995 1998 2001 2004

Percent graduating, P: 62.4 54.1 48.0 43.5 41.8

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

(b) Does d^2P/dt^2 appear to be positive or negative between 1992 and 2004?

(c) Explain why the values of P and dP/dt are troublesome to the principal?

(d) Explain why the sign of d^2P/dt^2 and the magnitude of dP/dt in the year 2001 may give the principal

some cause for optimism.

Answer 1

Joy

Question 11 since L=f(x) and the units are in min.then L'=f'(x) has to be in min/km, and f"(x) has to be in min/km²

^{Question 12 If you make a rough sketch of the data you will see that every 3 years there is a decline in P so the slope is negative}

^{b. the second derivative is appox. constant over the interval 1992 - 2004}

^{c. Because it shows declining graduation rates each year getting under 50%}

^{d.dP/dt appears to be slowing indicating the graduating rate might bottom out at about 40%}

^{I don't think that is a reason for optimism....}

^JIm