See the coversheet for instructions and the point value of each problem.
-
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}
-
Find the function \(f(x)\) for \(x \gt 0\) that has \(\displaystyle
f''(x)=x^{-2}\), \(f(1)=0\), and \(f(2)=0\).
- Evaluate the integrals:
- \(\displaystyle \int_{-2}^{3}(x^2-3) dx =\)
- \(\displaystyle \int_{1}^{4} \left(\frac{4+6u}{\sqrt{u}}\right)du =\)
- \(\displaystyle \int_{1}^{2}\left(\frac{x}{2}-\frac{2}{x}\right) dx =\)
- 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."
- \(\displaystyle\int_0^1 f(r)dr\)
- \(\displaystyle\int_0^3 f(x)dx\)
- \(\displaystyle\int_3^2 g(x)dx\)
- \(\displaystyle\int_1^2 (5f(x)+g(x))dx\)
- \(\displaystyle\int_0^1 f(x)g(x)dx\)
- \(\displaystyle\int_0^{14} f(x)dx - \int_2^{14} f(x)dx\)
- \(\displaystyle \int_0^2 f'(r) \; dr\)
- \(\displaystyle \int_6^6 f''(x) \; dx\)
- \(\displaystyle\displaystyle \lim_{x\rightarrow 1}\frac{f(x)}{g(x)-2}\)
- \(\displaystyle\displaystyle \lim_{h \to 0}\frac{f(2+h)-1}{h}\)
Last modified: Wed Nov 14 16:08:12 UTC 2018