# Midterm information/Midterm 2016/Commentary

### "When did we learn THAT?" and general comments

The list below provides a conceptual map of the midterm, indicating where the concepts underlying each problem appeared in the course.

1. Week 1. Asymptotics.
2. Week 2. This looks like a question about optimal foraging but it is really just about interpreting the shape of functions, in particular, Hill functions.
3. Week 4/5. Newton's method. This type of question appeared on Quiz 2 and in WeBWorK.
4. Week 5. This question requires thinking carefully about what it means to be an extremum. Examples of functions that show the statement is not true for all continuous functions: (a) $f(x)=x^3$. (b) $f(x)=|x|$. (c) $f(x)$ could have a maximum at $x=0$ and minima at $x=-1/2$ and $x=1/2$. (d) $f(x)=x^4$.
5. Week 3/4. Chain rule - of all the rules for differentiating (product, quotient, chain), the graphical version of the chain rule is the hardest. Write down the abstract formula for the chain rule and carefully evaluate the parts (approximations from the graph).
6. Week 2. This could also be done by recognizing that the limit has the form of a derivative with function $1/x$ and then using the power rule. This is less than ideal because of the conceptual circularity (one usually figures out how to calculate limits first and then finds out the power rule) and because we have not yet shown (in this course) that the power rule applies when the power is negative.
7. Week 4. No comment.
8. Week 4. Antiderivatives of polynomials. The derivative is a cubic function.
9. Week 5. This problem is a conceptual combination of the problem about Lucas and his father from the 2015 midterm (the circular portion of the track and distance to Mother) and OSH 2 (distance from a point to points along a straight line). The trickiest part is to realize that as Lucas' direction of travel turns from the straight away toward his mother, the distance between them starts to decrease more rapidly (second derivative negative) until the moment that he is instantaneously running straight toward her (inflection point). After that, he is turning away from her (his approach to her slows so concave up). From OSH 2, you should be aware that when you travel past a point laterally, the graph of distance to that point must be flat. The idea that the tangent line to a path travelled is the direction of motion came up in the space ship tangent line problem. Most of the points on this problem were not hard to get. The two points for determining concavity along the circular portion of the track were intended to be difficult and indicative of a likely A in the course. Getting this one AND #4 correct puts you the A+ range.