MT171: Calculus - Feedback on the January Test, 13 January 2016 Overall comment: when you write an expression or an integral you have to write a correct mathematical sentence. Also too many of you did not explain what you are doing. You are penalised for both of these! Feedback on individual questions: Q1: There were two separate steps here; calculating the vertical asymptotes by solving for when the denominator is 0, and solving for the slant asymptotes as x tends to both positive and negative infinity by factorising the polynomial. For solving when the denominator was 0, there were a number of common errors, including . Not noticing that a common factor x could be factorised (note: factorised, not cancelled), leaving a quadratic formula that has a standard solution, and instead trying to use Descartes rules of signs. . Claiming that the complex roots of the equation yield asymptotes . Not emphasising that x=0 is the only solution. The majority who chose to perform the polynomial factorisation did so correctly, but too many chose to handNwave and just worked out the leading order term. It was then necessary to emphasise the conclusion for taking the limit of y as x goes to both positive and negative infinity (most forgot the negative part). Q2: Part (a) was all or nothing - either you knew what you were doing, or had no idea. It is important at the start to say "Let y=arcsinh(x)". Bizarrely, many students seemed to equate arcsinh(x) with 1/sinh(x) which is absolutely not true. A few tried to fudge the answer by saying "Let u=x", claim an answer for dy/du and derive the fact that du/dx=1. This got no credit. In Part (b), provided students chose to take logs or, equivalently, write y=exp(x ln(arcsinh(x))), the use of chain rule typically followed without a problem. However, many students did not take this step and instead tried apply inappropriate rules of differentiation, such as claiming dy/dx=x (arcsinh(x))xN1. This rule only works if the thing that you're differentiating with respect to is not in the exponent. Q3: This was one of the better answered questions, although solutions could often have been laid out more clearly and logically. Using intermediate variables often helped with that clarity, but it is important to be very clear on what variable is being differentiated, and with respect to what. There were a number of slips or inconsistencies. Q4: This was by far the worst answered question. Many had no idea how to get started, perhaps trying to use partial fractions (which don't work due to the square root) or integration by parts. Even those who did complete the square often did so incorrectly (lack of care with the negative signs typically resulted in 7/4 instead of 9/4 as the remainder term), or, once the integral had been converted into a sufficiently standard form, the result was quoted instead of derived (even if you know the standard result, that should immediately tell you what the substitution is that