Complete the given table with the values of the 3 -unit moving average of the given function. HINT [See Example 3.] $$ \begin{array}{|c|c|c|c|c|c|c|c|c|} \hline \boldsymbol{x} & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline \boldsymbol{s}(\boldsymbol{x}) & 2 & 9 & 7 & 3 & 2 & 5 & 7 & 1 \\ \hline \bar{s}(\boldsymbol{x}) & & & & & & & & \\ \hline \end{array} $$

Variable Sales The value of your Chateau Petit Mont Blanc 1963 vintage burgundy is increasing continuously at an annual rate of \(40 \%\), and you have a supply of 1,000 bottles worth $$\$ 85$$ each at today's prices. In order to ensure a steady income, you have decided to sell your wine at a diminishing rate-starting at 500 bottles per year, and then decreasing this figure continuously at a fractional rate of \(100 \%\) per year. How much income (to the nearest dollar) can you expect to generate by this scheme? HINT [Use the formula for continuously compounded interest.]

Profit Your monthly profit on sales of Avocado Ice Cream is rising at an instantaneous rate of \(10 \%\) per month. If you currently make a profit of $\$ 15,000$ per month, find the differential equation describing your change in profit, and solve it to predict your monthly profits. HINT [See Example 3.]

Growth of Tumors The growth of tumors in animals can be modeled by the Gompertz equation: $$ \frac{d y}{d t}=-a y \ln \left(\frac{y}{b}\right) $$ where \(y\) is the size of a tumor, \(t\) is time, and \(a\) and \(b\) are constants that depend on the type of tumor and the units of measurement. a. Solve for \(y\) as a function of \(t\). b. If \(a=1, b=10\), and \(y(0)=5 \mathrm{~cm}^{3}\) (with \(t\) measured in days), find the specific solution and graph it.

Meteor Impacts The frequency of meteor impacts on earth can be modeled by $$ n(k)=\frac{1}{5.6997 k^{1.081}} $$ where \(n(k)=N^{\prime}(k)\), and \(N(k)\) is the average number of meteors of energy less than or equal to \(k\) megatons that will hit the earth in one year. \({ }^{49}\) (A small nuclear bomb releases on the order of one megaton of energy.) a. How many meteors of energy at least \(k=0.2\) hit the earth each year? b. Investigate and interpret the integral \(\int_{0}^{1} n(k) d k\).

Decide whether or not the given integral converges. If the integral converges, compute its value. $$\nabla_{-\infty}^{0} \frac{2 x}{x^{2}-1} d x$$