# Test 2 solutions and redo problems (bumped up to top of posts)

Here are the solutions to Test 2, according to the version you have (look at the lower left on the cover sheet)

MAT1575Test2a-solutions    version $\alpha$

MAT1575Test2b-solutions   version $\beta$

Here are some errors that were frequently made, which I did not mention in those solutions:

• The integral $\int \frac{\textrm{d}x}{x}$ is $\ln|x| + C$: the absolute value signs are essential! Too many people left them out. This can cause you to think a definite integral doesn’t converge when it really does. Be careful!

• The definitions of the improper integrals where there is a singularity of the integrand (the range is unbounded) involve one-sided limits: it is important to take the correct limit, because the two one-sided limits may not agree with each other and it is possible that one of them exists and the other does not. Please look at the definition given in the textbook on p. 337 (Definition 25) or see in Paul’s Online Notes: Improper Integrals (scroll down to “Discontinuous Integrand”) where there is more detail.

• Problem #6 specifically asked that you use a comparison test to determine whether or not the integral would converge. Some people just went ahead and tried to compute the integral. This was wrong (because it was not what was asked for) and a waste of time. Please read the instructions to the problems. For comparison tests, here are Paul’s Online Notes: Comparison Tests for Improper Integrals.

• In general, there were a number of test papers where many steps were missing and I am quite sure those steps could not be done in your heads.I do not know how this came to be, but please know that in the future I will have to make sure that people are not using extra sheets of paper and do not have their cell phones visible. I am very sorry to have to do this, for the majority who are honestly doing their best and following the instructions. Please read all the instructions on the cover sheet; you are responsible for following them. In particular, if I cannot see how you got your answer or any part of it, you may receive no credit: if in doubt, write it out!

It would not be a bad idea to look at both versions of the test solutions. Only two problems were actually different, and some were moved around.

If you did not receive full credit (a total of 40 points) for problems 5 and 6 on the third page, you have the opportunity to work similar problems to increase your score. In order for your score to increase, these must be done totally correctly and showing all work (except what you can easily do in your head: if in doubt, WRITE IT DOWN!)

These problems are due by the start of class on Wednesday 8 November.

MAT1575Test2-redoProblems