# MAT137: Calculus!

### University of Toronto 2017-2018

Test 2 took place on Friday, December 1st. Click here to see the Test 2 information page (as it looked before the test).

• We have emailed you an electronic copy of your graded paper to your "@mail.utoronto.ca" email address. It contains the score per page and a few comments. The email came from "Crowdmark Mailer (no-reply@crowdmark.com)" with a subject line like "Crowdmark - Test 2 - Marks Available".

## How to interpret the results

We always aim to create fair tests, and we think this test was fair. Your grade on this test will give you honest feedback on how well you meeting the learning objectives of this course. If you understood why it was necessary to note that the function in Q5 was positive and continuous in order to make sense of your algebraic manipulations and evaluate the limit, you likely have quite a good understanding of limits and continuity. Please also note that almost exactly the same question was included in Problem Set B. If you were able to draw a correct graph for Q8, you likely have quite a good understanding of inverse functions.

In particular, we think of Q7(b) as great problem for testing your understanding of several different things you have been taught in this course. The results for this question were broadly split into two categories:

• Students who understood very well what they were doing, and got a perfect or near-perfect score. Relatively few students understood the question but still scored poorly.
• Students who fundamentally misunderstand some aspect of the question, and who in most cases tried to prove that f(a) > f(b). An answer of this sort is cause for serious concern, since for example 1) this makes no sense, since f(a) and f(b) need not be defined; 2) this is not the definition of "decreasing" (a definition you were asked to write in Q6(c), note); 3) all of the instructors spent significant class time on a very similar proof. It is difficult for us to imagine a student who comes to and participates in lecture, and who engages seriously with the material, giving this sort of answer.
If you were able to write a correct proof for Q7(b)--which notably must involve starting with an arbitrary pair of points from the interval--you likely have a good understanding of how to use definitions and theorems, and what quantifiers mean.

Other common errors we saw during the grading include:

In Q4 (the optimization problem), many students failed to specify the domain on which they were trying to optimize their function. Many students also didn't correctly check whether they had found a global maximum, instead only justifying that they had found a local maximum.

In Q6, many students included many unnecessary hypotheses in their definitions. Notably, none of these definitions require an assumption that the function is continuous or differentiable at any point. Many students also used the word "let" when they should have been using "for all". For example, the definition of injectivitiy should begin with something like "For all pairs of distinct points in I...", rather than "Let a, b be distinct points in I. ..."