Jane finally made her way to Sixtopia to battle the infamous gym leader there, Pat Wa. He is feared for his vast knowledge of math and many have crumbled before him, cowering in fear whenever young trainers attempt to defeat him. Unfortunately there are the stubborn trainers that enter the gym with warning, but leave the building scarred with defeat and bloodshot eyes. For every clump coming in, there is always a group running out. The question is…will Jane come out running?

You are challenged by Leader Pat Wa!

Leader Pat Wa sent out Entreeng!

"Go! Gralinte!"

The foe used an INFORMATION!People are entering and leaving the gym at the same time 24 hours a day. The rate at which people enter is given by the function:

On the other hand, the rate at which people exit is given by:

The foe’s Entreeng’s defense has risen!

Jane used an ATK BOOST!

Gralinte’s attack has risen!

The foe's Entreeng used QUESTION!

(a) How many people have entered the Gym to at 6pm?

Gralinte flinched! Gralinte takes 5 damage!

Gralinte uses INTEGRATION!

It’s a critical hit! Entreeng takes 34 damage!

First we must take the integral of R(t) from 0 to 18 since we are only looking for how many people have entered at 6pm. Since people start entering at 0 hours, we can assume that 6pm will be equivalent to 18 (Simply think of it as following the 24 hour clock.

The foe’s Entreeng used INTIMIDATE!

Gralinte’s defense has lowered!

“Gralinte, that’s enough! Come back!

Go! Derivee!”

Derivee used ANTI DIFFERENTIATE!

Entreeng takes 16 damage!

Assuming the fact that we’re not allowed to use a calculator, we will need to integrate by hand. Remember the First Fundamental Theorem of Calculus, which states that the integral of a derivative = total change in the parent function. So anti differentiating R(t) should give us:

Which simplifies to:

The foe’s Entreeng used FEAR!

Derivee is unaffected!

Now that the expression is anti differentiated, we can now plug in the units and apply the fundamental theorem of calculus, which is to find the total change in the parent function along the interval. So substitute in the two interval values into the equation and subtract.

This leaves you with a calculated answer of 1296.3333 which means that approx. 1296 people have entered the gym at this time.

Derivee used SUBSTITUTION!

Its super effective! Entreeng takes 40 damage!

The foe's Entreeng fainted!

Derivee gained 120 EXP. Points! Derivee grew to LV. 6!

“You’re quite impressive! But can you beat my next Mathemon?”

Leader Pat Wa is about to send in Entorlev. Will you switch your Mathemon?

“Yes.Derivee, that’s enough! Come back!Go! Gralinte!”

Leader Pat Wa sent out Entorlev!

The foe's Entorlev used QUESTION!

(b) How many people are in the gym at 4 am?

Gralinte shudders! Gralinte takes 8 damage!

Gralinte used INTEGRATION!

Its super effective! Entorlev takes 36 damage!

In order to find how many people are in the gym at 4am, we need to integrate the difference between the rate at which people are entering and the rate at which people are leaving from 0 to 4.

Normally, you would be given a calculator to solve these types of questions, but we’ll go through the steps in anti differentiating the fraction only since the steps have already been shown in part a.

Entorlev used MIMICRY!

It has no affect on Gralinte!

“Gralinte, that’s enough! Come back!

Go! Derivee!”

Start by using the “u substitution” technique by taking the most complicated part of the fraction and letting it equal “u”.

Then take the derivative so that we can find a way to substitute out the dt, in order to make everything in terms of “u”.

Derivee used ANTI DIFFERENTIATE!REMINDER: We’re only showing the anti differentiation of the fraction in this part.

We can’t continue with this alone since not everything is in terms of “u”. As you can see, we still have a “t” left in the numerator. First let’s bring out the 15 so that we just have “t” at the top.

To get rid of the “t”, first remember in our first step that we made “u = 1 + 3t”. From here, we can solve for “t”, leaving us with an integral in terms of “u”.

Now substitute in the newly found value of “t”.

Simplify a bit..

We can now start anti differentiating the function in order to get our parent function to work with.

Let’s just take a look at the integral by itself and worry about the outside fraction later.

This is actually pretty easy to anti diffirentiate. The hardest part being the fraction since some confuse its derivative/anti derivative pairing.

Now we can bring our focus back to the whole thing.

Let’s just take a look at the integral by itself and worry about the outside fraction later.

This is actually pretty easy to anti diffirentiate. The hardest part being the fraction since some confuse its derivative/anti derivative pairing.

Let’s not forget the fraction we had on the outside of the integral.

And there’s your anti derivative! Now we can get back on track to the original question, which was to find how many people are in the park at 4 am. Remember what we found out originally:

REMINDER: We’re subtracting the exiting function from the entering one and integrating since that represents how many people are currently in the

gym.

Now let’s apply the fundamental theorem of calculus since we know the anti derivatives, or parent functions, of each.

Solving for that using a calculator, or if you have the time to do it by hand, we should come up with an answer of 48.6082. Since we are counting in terms of people, we will round down to 48 since you can’t have 48 and about a half people.

∴ There are 48 people in the gym at 4 am.

The foe's Entorlev fainted!

Derivee gains 192 EXP. Points! Derivee grew to LV. 10!

“Wow! I’ve never had anyone come this far before. Hupsha! Hupsha! Quick like a bunny! Let’s keep going, although I’m not sure if you can withstand this last one!”

Leader Pat Wa is about to send in Levorente. Will you switch your Mathemon?

“Yes.Derivee, that’s enough! Come back!Go! Gralinte!”

Leader Pat Wa sent out Levorente!

The foe’s Levorente used QUESTION!

c) Calculate the number of people in the gym at 8 pm if people stopped

entering at 3 pm, and then started entering again at 7 pm. NOTE: People are continuously leaving from 12 am to 8 pm.

Gralinte takes a critical hit! Gralinte takes 15 damage!

Gralinte used INTEGRATION!

This question is very similar to the last one. The only difference is that you may have to think of this question in terms of parts and then putting everything together.

Let’s take a look at the first part of the integration first and then try to make some sense of what is going on in that specific part.

Reading the question again, we are told that people are continuously entering from 12 am to 3pm and then from 7 pm to 8 pm. We are also told that people are continuously leaving along the whole time interval of 12 am to 8 pm.

Converting the times in terms of our interval, we’ll see that from 12 am to 3 pm is 0 to 15. Since there are still people entering, we end up with the integral above.

Moving along we have:

This part represents the period of time in which people are no longer entering the gym. Knowing this, we only need to integrate along the specific time interval and only the exiting function. We end up subtracting this integral from the previous one since this represents the people who have left the gym.

Lastly we have:

This integral represents the part at which trainers have resumed entering the gym, which is basically the same procedure as the first part but a different interval since this is the second period of time trainers are entering.

Also keep in mind that you are adding this to the previous two since this represents the total people in the specific time interval.

This question definitely wouldn’t require you to do manually. If for some strange reason that does happen though, you already know how to apply the fundamental theorem of calculus and we’ve already shown how to anti differentiate both functions.

So using a calculator we should come with an answer of approximately 963.1848. Like the previous question stated, we can’t have a decimal of a person, so let’s round down to 963.

∴ There are 963 people in the gym at 8 pm.

Gym Leader Pat Wa is a busy guy o_o

The foe's Levorente fainted!

Gralinte gained 234 EXP. Points! Derivee grew to LV. 8!

Jane defeated Leader Pat Wa!