AP Cheesecake: The Many Duels of Andrew Jackson

We now continue on our journey of working the 2010 AP Calculus questions with question number two.

2010 AP Calculus AB Free Response Questions

Our story now moves from John Quincy Adams's albino alligator Tim, to the many duels of Andrew Jackson.

As we all know, Andrew Jackson was a badass. Among his many achievements are:

- First president to be attacked while president
- First president to have an assassination attempt
- Survived 13 duels, injured many times, killed a man
- Stopped Civil War between Michigan and Ohio
- Destroyed the 2nd Bank of the United States
- Enlistment of pirates (technically privateers) during Battle of New Orleans

Dramatization

- Ironically ended up on this even though he directly caused the Panic of 1837:

1905 George Washington $20 bill was so much better

One fact that has been lost in history is that Jackson enjoyed cheesecake, especially before and after duels. In fact, he used to go on cheesecake binges before duels.

In one of Jackson's recently uncovered journals, he recorded the amount of cheesecake he consumed:

(a) Approximate the rate at which Andrew Jackson is consuming cheesecake, in hundreds of grams per hour, at t = 6.

Since, for some reason, Jackson decided to only record the amount of cheesecake he ate at odd intervals (probably because of the cheesecake induced stupor), we will have to "approximate", as the problem says, the rate at which Jackson is eating cheesecake.

The two tricks to realize here are that it's asking you for the rate, and it asks you to approximate. We have the chart of the amount, so in order to get the rate we will need the derivative, but since there's no way to take the derivative of this we will have to approximate it by using the slope between t = 5 and t = 7:

 Units are in hundreds of grams of cheesecake per hour because that is what the slope would be of a line created by a plot of hours vs hundreds of grams of cheesecake.

(b) Use a trapezoidal sum with four sub-intervals given by the table in order to approximate:

Using correct units, explain the meaning of this in terms of number of entrees.

This question tests your knowledge of how a trapezoidal Riemann Sum works and you knowledge of understanding how an integral works in this context.

 Remember that the formula for a trapezoid is:

In order to take the Riemann Sum, we must calculate the areas of the four rectangles the table in the problem forms:

Which adds up to around 10.688 hundred grams of cheesecake per hour. The 1/8 serves to indicate that this is an average of the amount of cheesecake consumed in hundreds of grams per hour from t = 0 to t = 8.


(c) At 8 P.M., Jackson began the painful process of throwing up cheesecake at a rate of:

If this occurs from t = 8 to t = 12, how much cheesecake is still inside Jackson at t = 12?

In order to solve this problem, you must realize that what you need to find is an amount, and what they give you is one amount and one rate. The amount of cheesecake Jackson has consumed at t = 8 is given in the chart, and the rate of expulsion of cheesecake must be subtracted from that. In order to get an amount of cheesecake, we must take the integral of the rate from t = 8 to t = 12 which is to be subtracted from the amount of cheesecake at t = 8:

(d) According to the model from part (c), at what time is Jackson upchucking cheesecake at the greatest rate?

To find the time at which cheesecake is being upchucked at the greatest rate, we must take the derivative of the rate which is given to us as T (t). Then we must find the local maximums, which is where T'(t) = 0 and is changing from increasing to decreasing. We must also remember that the maximum could also occur at either of the endpoints at t = 8 or t = 12:

 Therefore, t = 9.184 and t = 10.817. Now these times must be plugged into the original equation to find where the rate was the greatest. Remember to include t = 8 and t = 12:

Therefore, at t = 12, or midnight, is when Jackson was emptying the contents of his stomach at the fastest rate.

Questions? Concerns? Not enough Jackson? Contact me at Jonathan352@gmail.com.

Advanced Placement Cheesecake: Alligator Edition

Now that many of us have gone through the trauma and ordeal of taking the AP Calculus test, and it's been long enough for me to not jeopardize my AP scores, it is time to begin working the free response questions.

If you would like to find the original 2010 Calculus AB problems they are here:

2010 AP Calculus AB Free Response

Our first question will involve a certain alligator owned by John Quincy Adams:

 Sadly, it wasn't an albino alligator like this one

For convenience sake, the alligator will be from now on referred to as "Tim".

John Quincy Adams's alligator had a voracious appetite for cheesecake. There is no cheesecake in his food bowl before 12 P.M (t = 0). At precisely 12 P.M. John Quincy Adams begins to fill Tim's food bowl with cheesecake at a rate of:

Where t is in hours and f(t) is in cubic inches per hour.

Tim then begins eating the cheesecake at this rate:

 (a) How much cheesecake is in Tim's food bowl at 6 P.M.?

What the AP test writers are testing you over is whether you realize you need to take the integral of a rate to get the amount. Since from zero to six hours there is no cheesecake eaten and only cheesecake being added at a rate of f(t), all you have to do is take the definite integral of f(t) from zero to six:

Your units are feet cubed because the integral of a rate, inches cubed per hour, gets you the amount, which is inches cubed.

(b) Find the rate of change of cheesecake in Tim's food bowl at 8 A.M.

This question is testing your ability to realize that the question already gave you the rate and that you have to account for the addition of cheesecake and the removal of cheesecake. You should get something like this:

 

Units are inches cubed per hours because that is the rate given in the problem. The negative sign indicates that the amount of cheesecake is decreasing at that time. The negative sign is not necessary if you indicated that cheesecake was being removed or was decreasing at that rate.

(c) Let h(t) represent the total amount of cheesecake in cubic inches that Tim has eaten at t hours after noon. Express h as a piecewise function with domain zero to nine.

This is yet another problem testing your ability to recognize that you need to take the integral with a twist: this time you have to integrate a piecewise function. Don't fear though, all you have to do is integrate each separate part and then write it as three separate functions which should look something like this:

(d) How much cheesecake is in Tim's food bowl at 9 P.M.?

Yet again, more integrals, but this time you had to realize that you have to include the integrals of the rate of addition of cheesecake and subtract from that the integral of the consumption of cheesecake. This should look like:

The expression can be written in several different ways, but your answer should be the same no matter what.

Tim likes his cheesecake...

I will follow with the rest of the AP problems... soon.

If you have any questions or concerns about this problem, feel free to contact me at Jonathan352@gmail.com