1. AP Calculus Free Response Question
What to do?
AP Calculus Free Response Question
Caroline Cheung
Pd 2&3

2. 2010 - Question #2 t (hours) 2 5 7 8 E(t) (hundreds of entries) 4 132 5 7 8
E(t) (hundreds of entries) 4 13 21 23
A zoo sponsored a one-day contest to name a new baby elephant. Zoo visitors deposited entries in a special box
between noon (t = 0) and 8 P.M. (t = 8). The number of entries in the box t hours after noon is modeled by a
differentiable function E for 0≤t≤8. Values of E(t ), in hundreds of entries, at various times t are shown in
the table above.
The lesson today is…

3. At t = 6 there are 4 hundred entries per hour
Part A :
Use the data in the table to approximate the rate, in hundreds of entries per hour, at which entries were being
deposited at time t = 6. Show the computations that lead to your answer.
To find the rate at t = 6 you have to use the Mean Value Theorem:
According to what was given in the chart. Use t = 5 and t = 7 because 6 is between those two numbers.
E’(6) =
At t = 6 there are 4 hundred entries per hour

4. Part B: P t(hours) 2 5 7 8 E(t) (hundreds of entries) 4 13 21 23
Use a trapezoidal sum with the four subintervals given by the table
to approximate the value of
Using correct units, explain the meaning of
in terms of number of entries P
Is the average number of hundreds of entries in the box between noon and 8 P.M.
Ahhh!
t(hours) 2 5 7 8
E(t) (hundreds of entries) 4 13 21 23
The four subintervals are (0,2), (2,5), (5,7), (7,8)

5. Part B : Continue…
Note: The base would be the amount of entries added together and the height would be the difference between the t values.
Trapezoidal rule :
Giving you:
= or
PLUG IT IN!

6. Part C:
At 8 P.M., volunteers began to process the entries. They processed the entries at a rate modeled by the function P, where
hundreds of entries per hour for 8≤t≤12. According to the model, how many entries had not yet been processed by midnight (t = 12)?
It is given in the chart that at E(8) = 23.
Take the amount of entries processed which is E(8) subtract the integral of P(t) from 8≤t≤12. To determine how many entries were not processed.
???
Note: Integration is necessary because the function given models the rate.

7. Part C: Continue… Step 1: type in 23 –
Step 2: go to MATH then scroll down to the 9th one and press enter
Step 3: type in the equation
Step4: then press the following buttons: , X , 8 , 12
Step 5: press enter to solve.
= 7 hundred entries
Oh yeah. We got this.

8. Part D:
According to the model from part (c), at what time were the entries being processed most quickly? Justify your answer
The problem asked for what the entries are processed most quickly which means to find the maximum of the function.
Note: Find the derivative of the function in order to find the maximum of the function.
P’(t) = 0
To find the zeros of the function graph the function.

9. Graphing it. Plug it into the calculator to graph
Press 2nd trace and scroll down to ZERO.
Set the bounds and get t = and t =
The graph
That’s right

10. Part D: Continue
After obtaining the values plug it into the original equation (P(t)) to determine the maximum value. t
P(t) 8 12
At t = 12 the entries are processed most rapidly

11. THE END!
Let’s dance!

12. Works Cited