Still Know Your Calculus? Try The Tech’s 18.01 TestBrian Loux
Upperclassmen always enjoy taking freshmen naovete for outright stupidity. Is it an accurate description? While many have already mastered and passed the courses freshmen have yet to be taught, have we actually retained our knowledge from the past?
When the opportunity to pit myself against the freshmen arose, I eagerly decided to find out.
Yesterday morning, some members of the Class of 2006 awoke at 8:00 a.m. probably the last time for the next nine months to take the 18.01 advanced standing exam. I figured that I, a junior with the benefit of two years of MIT education, should be able to fare considerably well on a basic calculus exam.
I opted to take the 18.01 (Calculus) test instead of the 8.02 (Electricity and Magnetism) test because presumably all scientists and engineers use basic calculus on a fairly regular basis.
I wanted to be entirely unprepared for the test as well, an attempt to make the exam a random test of an MIT student's abilities. Thus I chose not to study for the test and got set for only five hours of sleep.
I'm a freshman, really
I realized going in that I had everything to lose. My own dignity, the reputation of The Tech, Environmental Engineers, the lacrosse team, Next House, the Freshman Urban Program, and most importantly the respect of Nora Buchanan, who taught me this stuff back in my senior year of high school.
After a long string of personal foolish mishaps, such as locking myself out of my dorm room, going to the wrong test room, and forgetting a pencil, I finally got into 10-250 half an hour late. I was nervous that today wasn't going to be the day to test my intelligence.
For full effect, I wore my South Lakes High School Seniors shirt to make my freshman appearance all the more convincing. The overseer eagerly accepted my apology about tardiness (I didn't even have to use my excuse about being at a mandatory event) and let me sit in the fourth row. I felt pleased to appear so young. Miraculously, the graduate student who asked to see my ID Card did not notice that it expired in 2004 and not 2006. Thus I was able to finish the test without incident.
A big boost of confidence hit me after I breezed through the first two pages of the test. I was halfway done in half the time allotted to the freshmen! When a tough problem arose, I found that most of my time was spent recalling classes from BC calculus, not really applying examples that I had learned.
Of course disaster had to hit. On the final page, I came to the realization that I couldn't do the first thing with integrals. I could work with approximations, but anything with more than two x's threw me for a loop. It was panic time. I tried to go back to the first page and think back to what implicit differentiation meant, but of course to no avail.
By this time, the really smart people were turning in their papers. There is nothing more detrimental to a test-taker than someone clomping to the front of the room and handing in his or her paper, reminding you that you must be stupid for not yet finishing. I kept telling myself that I arrived late and wasn't even taking the test for credit, but I couldn't keep focused. Stupid smart freshmen.
What impressed me the most was the confidence that freshmen seemed to display during their tests. Here they were for their first real MIT test in an intimidating lecture hall and already under an extreme amount of duress, but none of them showed the slightest bit of quivering. I commend the freshmen for the job they did and hope that their nerves stay for the rest of their years.
I felt confident when I finished the test. Botching three of 15 questions still gives me around an 80 percent, so that's still above passing. And I'm on pass/fail, right?
How smart is you?
As a researcher, I felt a tad unsatisfied from the test. How helpful is just one person's experience? So recalling some of the problems that I just faced, I wrote up The Tech's own calculus quiz and took it to the streets.
I was actually also hoping to find a mathematics professor and grade him really harshly, but with classes a week away, it was a tougher job than I expected. Getting students to take a math test was no picnic, either. I frequently heard the phrase "I'm no math major!" as I walked around campus. "I don't want to do it," said one senior. "I did better with a high school style teaching, and I think that's why I had a problem with the math classes I took here." Fellow Tech staffers even refused to take the quiz knowing that I couldn't print their results.
Others wanted to try and skew the results. "Go speak to somebody on Fourth West! They'll know it!" said an East Campus resident.
I was quite surprised to see how often people failed to remember the most basic elements and rules of the subject. It was almost a replica of the "don't drink and derive" t-shirt with a few other errors they forgot. Usually, a test-taker was a well-oiled integrating machine or a very rusty one. Others had lost their batteries completely.
While polling was entertaining, I wasn't really able to draw any conclusions about retention or class intelligence from the test. All I can say is I felt a lot better when I was no longer in the hot seat and became the 18.01 sage (because I had memorized the answers). I also felt like I was doing a public service, as many people made a vow to brush up on their math skills for next term.Need some paper?
These questions were modified from ones that were given on Wednesday morning’s 18.01 (Calculus) advanced standing exam. Participants’ answers are direct quotes made while writing out or verbally explaining the problem. To protect the dignity of some participants in this survey, the interviewed were able to classify themselves in any way they chose, though they had to provide their year. Questions are weighted evenly, with sub-questions given 10 percent apiece. Arithmetic errors were given half credit. The real 18.01 test required students to receive a grade of 70 percent or higher to pass out of the class.
For a challenge for the readers, answers are not listed, though solutions and solution methods can be discerned from the responses.
1) Find the maximum possible area of a rectangle that could fit in the space between the x-axis and the parabola -2x4 + 8.
2) No paper needed. What is the value of the limit as R approaches zero of (sin(pi + R) - sin(pi))/R?
3) Differentiate the following:
b) x3 / (x4 + 2x2 +4)\
4) Using implicit differentiation, find the slope at (1,1) on the curve y3x2 + x2 - 2y2 = 0.
5) Integrate the following:
a) sin (5x + pi)
b) x2 / (x3 +2)2
Two graduate students, French
(Note: The two students conversed in French and showed me the answers after finishing)
1) One argued with the other not understanding the method he was using. Eventually, he was won over. (Correct)
3a, b) (Correct)
5a, b) (Correct, though they complained that it was a difficult problem.)
Sophomore, Female, Asian
1) OK, for this you have to take the Reimann or Rye-man sum, either that or you take the integral of the parabola. No? (Incorrect)
2) No, I can’t do that one. (Incorrect)
3a, b) Fuck, I don’t care. That’s too much work. (Incorrect)
4) Differentiate each segment and place them together? That takes too long. (Incorrect)
5a) Sin of 5x plus pi times -- no, divided by five. And a negative sign. No! Cosine! I eventually got it! (Partial credit)
5b) Oh man. (Question changed to identify the method needed) What method? I don’t know that. (Question changed to what term would be needed for substitution) What term? OH! u! That would be x3 + 2. (Partial credit)
Junior, Next House
1) Draw the parabola ... ok, then write a function f(x) ... g(x), which would be the function squared ... ok give me some paper. (Draws a positive parabola) Damn you. (Draws a negative parabola) Hell, the hard part is writing the function here. (Incorrect)
2) That’s the cosine of pi, so that’s negative one. (Correct)
3a) Oh, these are evil. (Writes answer) This is a chain rule problem. (Correct)
3b) Well, I only know the product rule so I’ll do it the hard way. (Correct)
4) (derives equation, pauses to recall to factor out dy/dx for each term) so now I can just plug in the coordinates and y prime is negative four. (Correct)
5a) So this would be ... what? I forgot the plus C? (Incorrect)
5b) OK, well you have to do substitution. Now I don’t remember how to do it. Let me keep going. Oh, now the problem is simple. And I replace the u ... that’s it. (Correct)
Junior, Stupid, Iowan
1. x4? Well, that’s not really a parabola then. Well, you would use a set of parametric equations and then take the derivative of each. (Incorrect)
2. Let’s see ... that’s one ... shoot! Cosine of pi is negative! (Arithmetic error)
(Test cut short due to other time commitments)
Junior East Campus
1. Ok, the area of the rectangle will be two times x times the parabola’s y value in terms of x. So take the derivative of that and the critical value will be the maximum x value. Plug that back into the original equation. (Correct)
2. Man. Well, since you can’t have a zero denominator ... wait (friend tries to steer him in the right direction). Nuts, I don’t know. OH MAN! It’s a little disheartening that something so obvious isn’t recognizable after so short a time. (Incorrect)
3a) Chain rule. Simple. (Correct)
3b) Quotient rule. I remember that one. Denominator derivative times the numerator minus the opposite of that ... over the denominator squared. (Correct)
4) Yeah, it’s basically the chain rule and you have to factor out a dy/dx term each time and you’re ok. (Correct)
5a) Hmm, pretty much the reverse chain rule. Is it indefinite? Then add a constant. (Correct)
5b) That’s just substitution. u would be x3 + 2 and we get rid of the numerator through du. (Correct)
1) Easy. You plot the two on the X-Y coordinate system and take the line ... draw the two ... Wow, this has been a while. It’s almost embarrassing. Take the integral between the two lines? Ok, I don’t know. (Incorrect)
2) Either simplify it first or use l’HÔpital’s rule. Hmm, doesn’t it look like the definition of a derivative? (Incorrect, but close)
3a) OK, gimme a sec. You can see that I don’t do these on a daily basis. How’s that? Oh, I forgot the exponent! (Arithmetic error)
3b) All right, I’ll be more careful on this one. I’m not gonna remember the quotient rule so I’ll do the product rule instead. (Correct)
4) I don’t know what implicit differentiation is! I think I’d try to graph the function to get it’s tangent line. (Incorrect)
5a) Wow. This would have been pretty easy freshmen year. Jeez. Junior’s not that old. Ahh, negative cosine! (Arithmetic error)
5b) OK, tell the freshmen to write down as much as they can and confuse the graders and get partial credit. (Incorrect)
Compiled by Brian Loux