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}