The choice of problems on optimization is ok. Some of the integrals are basically the same, so I chose the most representative and disregarded the rest. I am also missing a few topics.

• There are three kinds of integrals by substitution, but you only chose one.
• We worked many different problems of the application of optimization to marginal analysis in class. I will ask a few of those in the final.

Also, I don't recall doing any problem like number 3, where you mess with high degree polynomials.

1. Find any local min or max of the function \( f(x) = x^2-8x+4 \) using the first derivative test.
2. The function \( f(x)=x^4-3x^3+5 \) has a critical point at \( x=1 \). Use the second derivative test to identify it as a local max or local min.
3. Find any critical points of the function \( f(x) = 6x^5 + 33x^4 -30x+100 \).
4. Find any inflection points of the function \( f(x) = 2x^3+3x^2-180x+3 \).
5. Find constants \( a \) and \( b \) so the minimum of the parabola \( f(x) = x^2+ax+b \) is at the point \( (6,2) \).
6. Find the global maximum and minimum of the function \( f(x) = 2x^3-3x^2-12x+3 \) on the interval \( [-2, 4] \).
7. Let \( C(q) \) represent the cost and \( R(q) \) represent the revenue, in dollars, of production of \( q \) items.
   • If \( C(50)=4300 \) and \( C’(50)=24 \), estimate \( C(52) \).
   • If \( C’(50)=24 \) and \( R’(50)=35 \), approximately how much profit is earned by the 51st item?
   • If \( C’(100)=38 \) and \( R’(100)=35 \), should the company produce de 101st item? Why or why not?
8. Find the following antiderivatives:

\begin{align} &\int x^5+x^{100}\, dx \\ &\int \frac{3}{x^{24}} + 3x^{-4} - 27x^3\, dx \\ &\int e^x - \frac{24}{x^{3/4}}\, dx \\ &\int \frac{x^2}{x^3+1}\, dx\\ &\int (x+1)\big( \tfrac{x^2}{2}+x \big)\, dx \end{align}