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. You can 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.
   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 $\sqrt{x}$ into, then $x_0=4$ would also get you a point.
11. The points here are for getting the horizontal and vertical coordinates of the min/max in the right location ($x_{max}$ should be between -40 and -24 and $N(x_{max})$ should be between 4 and 6). If you made it clear that the location of the IP of $N(x)$ lined up with the max of $N'(x)$ you could get that point even if it was outside the -40 to -20 range. 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 graph does not have to be correct - even a piecewise linear graph could get full points if all other details are correct.