Chesecake: Now With 100% More Calculus, Derivatives, and Tea

By now, most of you are probably thinking "Yummy... cheesecake..." and a smaller percentage are thinking "Okay... so where is the Calculus?"

Right here...

This leads us to one of the two major Calculus concepts:

DERIVATIVES

This is one of the words that you should ingrain into your mind as much as 'cheesecake'. That's right, this words is JUST AS IMPORTANT AS CHEESECAKE.

 = 

That's right... Cheesecake + Tiramisu = Calculus

 What a derivative can tell you, if you know how much of 'something' there is as time passes, is how the amount of 'something' is changing at any point in time. This is otherwise known as the instantaneous rate of change, or slope. 

Derivatives can be represented on a graph with tangent lines, which are indicated in red on the above graph. 

Now, there are two ways you can do a derivative of a function. One way is to plug your function into this:

Warning: May lead to brain implosion

Note: f '(x) is denotes that it is the derivative of the function f (x)

If you remember what the definition of slope is, then the definition of a derivative may look slightly familiar:

 Slope equation

The numerator part of the derivative definition gives you two 'y' points or your 'rise' and the denominator 'h' is the difference between the 'x' points giving you your 'run'. So, therefore you have your rise over run or the definition of slope. Remember that slope also means the rate at which something is happening.

No need to be afraid of the derivative though! There is a shortcut to do your derivatives so you don't have to have six lines of work in order to derive x² (as indicated at the beginning of this post). 

One of the most common saving graces is called the:

POWER RULE 

Black... er... POWER RULE!
  

 There are many other shortcuts, but this one is by far the most used and easiest to perform. What you can do is multiply the coefficient by the exponent and then lower the exponent's power by one.

This can be mathematically represented by:

f '(x) = ax^(a-1)

So, instead of taking up six lines and slowly corroding your brain with the definition of a derivative in order to derive x², you can use the power rule to say it is 2x.

Some other examples are:

f(x) = x³

f’(x) = 3x²

f(x) = 2x⁴

f’(x) = 8x³

^{Since you can take the derivative of these functions, you can now plug in any 'x' value into the derivative function to find the slope at that point in the graph, which is also the change in rate at that point.}

NOTE: The power rule is only applicable with isolated terms with one variable. For example, you cannot use the power rule on (x+y)^2 or x*y.

^{If you are still confused on exactly what a derivative is, this example problem may help:}

^{Boston Tea Party}

 ^{342 chests of tea on a boat... 342 chests... take one down, throw it around... 341 chests of tea...}

^{On the night of December 16th, 1773, a group of men dressed as Mohawk Native Americans headed towards the Boston Harbor and dumped all the tea off three ships. The amount of tea dumped off the ships can be modeled as:}

 ^{f(t) = .01 * t^2 where 0 min < t < 185 min}

^{Where 't' is the time in minutes and f(t) is the number of crates dumped off of the ships.}

^{(a) Take the derivative of f(t) and explain what f '(t) means in context of the problem.}

^{(b) Find how fast tea is being dumped at t = 100 min. Be sure to indicate units.}

<sup>(c) Find how fast tea is being dumped after 144 crates have already been dumped. Be sure to indicate units.

Solutions:

For some reason, most teachers don't explain what the purpose of the problem was or how the problem was created. This is similar to asking someone to make a cheesecake without the knowledge of what the ingredients are. You may have the instructions on the process of making a cheesecake (just as you may know how to do a derivative) but not knowing the ingredients (or the makeup of the Calculus problem) can make making a cheesecake (or doing a Calculus problem) much more difficult than it should be.

The reason why the bounds for the problem are 0 min < t < 185 min is because you can't have negative time and if you plug in 185 minutes for t into f(t) then you get around 342 chests of tea, which was the number of chests of tea that the Sons of Liberty dumped on that day.

(a) f '(t) = .02t</sup>

<sup>This was asked in order to test your knowledge of taking the derivative of a function. For this, you could have used the definition of a derivative (the hard way), or used the power rule (the easy way).

For the power rule, you multiply the coefficient, which in this case is .01, by the exponent, which in this case is 2, and then subtract one from the exponent. Thus, you get f '(t) = .02t

The meaning of the derivative is the rate at which tea is being dumped at certain points in time. Since the units are crates of tea and time in minutes, then the unit that represents this derivative is crates / minute.

(b) 2 crates / minute

This was asked in order to see whether you realized that "how fast" means the rate at which tea is being dumped. Since the derivative of a function that indicates an amount (in this case, the total amount of tea that is dumped at a certain period of time) is the rate that the action is taking place (in this case, how fast they are dumping the tea at a certain period in time).

To solve this, all you have to do is plug in the 100 min. into your derivative. Therefore, at 100 minutes, they are dumping tea at around 2 crates per minute. As you can see, the units include 'per minute' which tells you the rate of an action.

 f '(100) = .02 (100) = 2 crates / minute

(c) 2.4 crates / minute

This problem tested your ability to understand the relationship between the derivative and the original function. Yet again, you needed to understand that "how fast" means the rate at which tea is being dumped, but you also needed to understand that the information it gave you was not a time, but the amount of crates that had been dumped.

This is a two step problem. You had to first find out the time at which there had been 144 crates of tea dumped and then plugged in that time into your derivative in order to find out the rate at which tea was being dumped at that time.

144 = .01 * t^2 
t =  120 minutes

This is done in order to find out the time at which 144 crates of tea had been dumped.

f '(120) = .02 (120) = 2.4 crates / minute

Then you can plug in 120 minutes into your derivative to find the rate at which tea is being dumped per minute when 144 crates were dumped.

If you have any questions or complaints over any of this post, feel free to e-mail me at Jonathan352@gmail.com or post in the comments.

Also, saying that the example problem should have been about cheesecake instead of tea is a silly complaint. It is common knowledge that tea is on the same level as cheesecake. Just look it up in the Fact Book of Facts.</sup>