MATH 2301-102 and -103 Fall 2017 Calculus I Recitation 12 Week 14

See the coversheet for instructions and the point value of each problem.

1. 4-methylcyclohexanemethanol (MCHM) leaked from a tank at a rate of $$r(t)$$ liters ($$\mathrm{l}$$) per hour ($$\mathrm{h}$$). The rate decreased as time passed and the values of the rate at two-hour intervals are shown in the table. Find lower and upper estimates for the total amount of MCHM that leaked out. What is the integral that expresses this amount? \begin{array}{|c|c|c|c|c|c|c|} \hline t (\mathrm{h}) &0&2&4&6&8&10\\ \hline r(t) (\mathrm{l/h}) &8.7&7.7&6.8&6.2&5.8&5.3\\ \hline \end{array}
2. Suppose $$f$$ and $$g$$ are differentiable functions with the following properties: $\begin{array}{lll} f(0)=2 &f(1)=0 &f(2)=1 \\ g(0)=1 &g(1)=2 &g(2)=0 \\ \int_0^1 f(x)dx=\pi &\int_1^2 f(x)dx=\pi^3 &\int_2^3 f(x)dx=\pi^5 \\ \int_0^1 g(x)dx=\sqrt{2} &\int_1^2 g(x)dx= \sqrt{3} &\int_2^3 g(x)dx=\sqrt{5}\\ f'(0)=e &f'(1)=e^3 &f'(2)=e^5 \\ g'(0)=\sqrt{7} &g'(1)=\sqrt{11} &g'(2)=\sqrt{13} \end{array}$ Evaluate the following. If one cannot be evaluated with the given information, write "NOT ENOUGH INFORMATION."
1. $$\displaystyle\int_0^1 f(r)dr$$
2. $$\displaystyle\int_0^3 f(x)dx$$
3. $$\displaystyle\int_3^2 g(x)dx$$
4. $$\displaystyle\int_1^2 (5f(x)+g(x))dx$$
5. $$\displaystyle\int_0^1 f(x)g(x)dx$$
6. $$\displaystyle\int_0^{14} f(x)dx - \int_2^{14} f(x)dx$$
7. $$\displaystyle \int_0^2 f'(r) \; dr$$
8. $$\displaystyle \int_6^6 f''(x) \; dx$$
9. $$\displaystyle\displaystyle \lim_{x\rightarrow 1}\frac{f(x)}{g(x)-2}$$
10. $$\displaystyle\displaystyle \lim_{h \to 0}\frac{f(2+h)-1}{h}$$
3. Evaluate the integrals:
1. $$\displaystyle \int_{-2}^{3}(x^2-3) dx =$$
2. $$\displaystyle \int_{1}^{4} \left(\frac{4+6u}{\sqrt{u}}\right)du =$$
3. $$\displaystyle \int_{1}^{2}\left(\frac{x}{2}-\frac{2}{x}\right) dx =$$
4. $$\displaystyle \int \frac{\sin(x)}{1-\sin^2(x)}dx =$$