Work in a group of at most 4. Explain to those who do not understand. Ask questions if you do not understand.
- (10 points) Use differentials to approximate the value of \(\sqrt[3]{26.8}\) by hand.
- (10 points)
What is the propagated error in the measurement of the cross sectional area of a circular log if the diameter is measured at 24 cm, accurate to 0.2 cm?
- (20 points) Find each of the antiderivatives. (Remember the "\(+C\)"!)
- \(\int 2\,dx=\)
- \(\int x^2 \,dx=\)
- \(\int \frac{1}{x^2}\,dx=\)
- \(\int \frac{1}{x}\,dx=\)
- \(\int x^{1/2}\,dx=\)
- \(\int \cos(x)\,dx=\)
- \(\int \cos(7)\,dx=\)
- \(\int e^x\,dx=\)
- \(\int x^2+1\,dx=\)
- \(\int \frac{1}{x^2+1}\,dx=\)
- (10 points) Find the function \(f(\theta)\) that has \(f''(\theta)=\sin(\theta)\), \(f'(\pi)=6\), and \(f(\pi)=5\).
- (25 points)
For the function \(\displaystyle f(x)= \frac{x-1}{x}\):
- Find any vertical asymptotes.
- Find any horizontal asymptotes.
- Find the intervals on which
\(f\) is increasing or decreasing.
- Find the local maximum and minimum values of
\(f\).
- Find the intervals of concavity and the inflection points.
- Use the information above to sketch the graph.
- (25 points)
A cone-shaped drinking cup is made from a circular piece of
paper of radius \(5\,\mathrm{in}\) by cutting out a sector and
joining the edges \(CA\) and \(CB\). Find the maximum capacity
of such a cup.
Last modified: Fri Mar 25 18:25:31 UTC 2022