Sections 11 and 12

Really cool review exam. Nice balance between intermediate and challenging problems. Lots of fun.
I am glad you decided to include some trigonometry.
I am missing some more variety of limits, though. Don't forget studying for all those!
Except for that last item, and the maybe excessive difficulty of some problems (which I really enjoyed), I believe this could be a good exponent of what the corresponding part of my final will be. Good job!

1. Sketch the graph of the following functions. You must find zeros, domain, range, vertical and horizontal asymptotes, intervals of increase/decrease, and intervals of concavity. Indicate also local extrema and inflection points.
   \begin{align} f(x) &= 4x^2+24x+32 \\ f(x) &= \frac{x}{x-24} \end{align}
2. Find the derivative of \( y \) using logarithmic differentiation:
   \begin{equation} y = \frac{25^{x+3}\sin x}{(x^2+4)\sqrt{x^3+8}} \end{equation}
3. Find the critical points of \( f(x) = x^{1/3} - x^{-2/3} \).
4. Compute the following limits:
   \begin{align} &\lim_{x\to \infty} e^{6x}x^{-1/2} \\ &\lim_{x\to \infty} \frac{23x-14x^3+7}{4x^2+7} \end{align}
5. Find all critical points of the function \( f(x) = \dfrac{x-4}{x^2-2x+8} \),
6. Find the absolute extrema of the following functions, in the indicated intervals:
   \begin{align} f(t) &= 2\cos t + \sin 2t && [0, \tfrac{\pi}{2}] \\ f(x) &= xe^{-x^2/8} && [-1, 4] \end{align}

Sections 15 and 16

We have not covered Mean Value Theorem in this class! Why do I see 4 problems on that topic? Two of those problems are actually extremely challenging, and belong in a much more advanced class. I am not happy about that. Of course, none of the corresponding problems made it to this review exam, and the students that wrote them will get no extra credit for it.
One of the optimization problems did not make any sense. I did not include it in the list.
All the problems regarding critical points are basically the same function. I wish you had put a little bit of more effort into looking at the homework, for instance, and chosen more relevant and challenging questions.
Not all the limits made it to this test either. I am missing many kinds of limits that can be solved with L'Hopital techniques, too.
All in all, I am sad to say that this exam is not very representative of what we have done in this part of the course, and pretty much doubt working on it will prepare you for the final.

1. Find the absolute extrema of the following functions, in the indicated intervals:
   \begin{align} f(x) &= x^3 -3x^2 + 1 &&[-\tfrac{1}{2}, 4] \\ f(x) &= \frac{x}{x^2+1} &&[0, 2] \\ \end{align}
2. Use any of the derivative tests to find local extrema of the function \( f(x) = x^4 - 4x^3 \).
3. Compute the following limits:
   \begin{align} &\lim_{x \to 0} \frac{e^x-1-x}{x^2} \\ &\lim_{x\to 0} (1-2x)^{1/x} \\ &\lim_{x\to \infty} \frac{e^{6x}+5}{2x^3} \\ \end{align}
4. Sketch the graph of the following functions:
   \begin{align} f(x) &= \frac{x-1}{x^2} \\ f(x) &= x^3 + 6x^2 + 9x \\ f(x) &= \frac{x^2}{x^2+9} \end{align}