Midterm information/Midterm 2015/Commentary

Here are some clarifications of the marking scheme for the midterm that are not included in the pdf file.

9. 1. It is possible to arrive at the correct answer (3/2) using an incorrect formula for the linear approximation. The incorrect formula that "works" is $f(2)\approxf'(2)+f(2)(2-1)$. This would get you 1 point simply for having the right idea for how to proceed even though you got the formula wrong. If you got a point for an answer of 3/2 by this incorrect method, DO NOT get the impression that you got the point for getting the numerical answer right.
   2. Mentioning concave up is not sufficient to get a point. Concavity must be linked to the approximation being an underestimate to get the first point.
10. Providing a function $g(x)$ that has a root in the correct location ($25^{2/3}$) gets you a single point unless it is a polynomial (polynomials must have INTEGER powers). Stating $x_0=3$ should get you a point for any $g(x)$. If your $g(x)$ included $\sqrt{x}$ somewhere in it, then $x_0=4$ would also get you a point.
11. You get two points for getting the horizontal and vertical coordinates of the min/max in the right location (one point for each coordinate). $x_{max}$ should be between -40 and -24 and $N(x_{max})$ should be between 4 and 6. There was also a point for showing $N'(0)=0$. Another point could be removed if your graph came down to zero too early. The slope should not hit zero above x=-80 nor below x=80. The concavity of the $N'(x)$ does not have to be correct and is difficult to discern from $N(x)$ - even a piecewise linear graph could get full points if all other details are correct.
12. The graph should start above zero and should be smooth (no corners) and flat at $t=0$, $t=1$ and $t=2$. It should not consist of straight lines. The explanations should make reference to Lucas and his father, not the graph. Simply stating what is effectively a definition of a min, max or IP is not enough. The crucial facts that you should demonstrate awareness of are that (1) whenever Lucas' directions of travel (tangent line) is perpendicular to the line between him and his father, he is neither approaching nor moving away from his father and so the graph must have a zero slope and (2) whenever Lucas' directions of travel (tangent line) is parallel to the line between him and his father, he is moving toward or away his father at a maximal rate and hence an inflection point.