Blogs, Calculus, Mathematical Reasoning

Exam: How much should you study?

Finals are approaching ever faster, and I wanted to give my students a little extra review for their Calculus final.  In so doing, I came up with the following questions to help them study and motivate them to study at the same time.  Whether or not you have finals coming up or not, can you answer these questions?

Change in grade from studying. (Related Rates)

In general, the more time you spend studying, the better you will do on an exam (we will revisit cases where this may not be true in the future).  However, for this problem assume that the grade, \(G\), you receive as a percentage on your exam can be modeled by the function \(G(t)=100-60e^{-.2t}\) where \(t\) is time spent studying in hours.

How long will you have to study in order to get an A \((\geq 90)\) on your exam?

If you have been studying for 4 hours, what at what rate is your expected grade changing with respect to the amount of time studying?

As you spend more time studying, does the rate of change of \(G\) with respect to time increase or decrease?

Sleep or Study? (Optimization)

Suppose that it is 8 PM the night before you final exam.  Your exam starts at 8 AM.  Your grade will be determined by the function above if you get exactly 8 hours of sleep.  However, if you do not get an optimal amount of sleep it will affect your mental faculties and you will receive a lower grade.  Assuming that the affect of sleeping on your grade, \(GS\), can be determined based on the time spent sleeping, \(s\), by the equation \(GS(s)=-5(8-s)^{2}\).

You then have that your grade as determining by time spent studying, \(t\), and time spent sleeping, \(s\), is given by \[G(s,t)=100-60e^{-.2t}-5(8-s)^{2}.\]  Assuming that you will only be sleeping and studying until the exam begins, how much time should you spend doing each in order to receive an optimal grade on your exam?  What is the maximum grade you will receive?

How long for each question? (Optimization)

As you’re taking your exam, you work your way through the problems and do the best that you can.  As your are nearing the end of the exam, you hear your professor state that there are 10 minutes left until the end of the exam.  However, you still have 2 problems left.  You don’t actually know how to do either one completely, but you know enough to be able to get partial credit on the problems if given time to complete them.

The first problem, you know a little better, and the number of points you will get, \(P_{1}\), based on the time you spend on the problem, \(t_{1}\), is given by \(P_{1}(t_{1})=10-10e^{-.23t_{1}}\).  For the second problem, the number of points you will get on the problem, \(P_{2}\), based on the time you spend on the problem, \(t_{2}\), is given by \(P_{2}(t_{2})=10-10e^{-.16t_{2}}\).  

How much time should you spend on each of the problems in order to maximize the sum of the two scores?

Is cheating the answer? (Definite Integrals and Expected Outcome)

Suppose that going into your final exam, your probability of getting a certain grade in a class can be determined by \[p(x)=\begin{cases} \frac{1}{49}(x-80) \text{ if } 80 \leq x \leq 87 \\ \frac{-1}{49}(x-94) \text{ if } 87 < x \leq 94 \\0 \text{ otherwise.} \end{cases}\]

On the other hand, you have a system for cheating guaranteeing that you will receive a 100 on your final and therefore a 94 in the class.  However, there is a 40% chance that you will be caught and receive a 0 in the class (along with other repercussions).

What is your expected grade if you do not cheat?  What is your expected grade if you do cheat?  Which should you choose?

Heading to class (Related Rates)

Your are running behind trying to make it to your final.  As such, you are driving eastward along a road and are currently \(\frac{1}{2}\) a mile west from an intersection driving 55 mph in a 45 mph zone.  As you look off to your left, you notice a police car parked \(\frac{\sqrt{3}}{2}\) miles south of the intersection with his radar gun at the ready trying to catch people speeding. 

What does the radar gun clock your speed as?  Will you get a ticket for speeding?


Determine if the speed would be registered higher or lower than your previous answer if,

  • The police car was closer to (further from) the intersection.
  • The police car was moving toward (away from) the intersection.
  • Are there any observations to be made about what happens when police are trying to catch people speeding?

Good luck!

If you have final exams coming up, I wish you the best of luck.  I sincerely hope you get the grade you want and do as well as possible in class.  Hopefully you’ve seen that it is best not to put studying off to the last minute so that you can get a full night’s sleep before your exams, that you should be careful of time management during the exam, and that you shouldn’t cheat.  Even though you got away with speeding in the last problem, I would suggest giving yourself enough time to get to class without speeding, because the police officer will likely be close enough to the intersection to get you for speeding.

If you don’t have finals coming up and you are just reading along for the challenge and entertainment, thank you for doing so.  I hope the questions lived up to your expectations and that you will follow STEM and leaf through the follow button in the side bar or through Social Media.  While here, try our other quizzes, or visit some of our affiliates.

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