Suppose the population P of a certain species of fish depends on the number x (in hundreds) of a smaller kind of fish that serves as its food supply, where $P(x)=2 x^{2}+1$. Suppose, also, that the number x (in hundreds) of the smaller species of fish depends on the amount a (in appropriate units) of its food supply, a kind of plankton, where x=f(a)=3a+2. Find $(P \cdot f)(a)$, the relationship between the population P of the large fish and the amount a of plankton available.

A survey question was, "Do you favor or oppose scientific experimentation on the cloning of human beings?" Suppose an agency that advocated cloning of human beings wanted to conduct a survey that would show support for their cause. Explain how it might use each of the following to do so:

Selection bias.

Suppose the demand for a certain brand of vacuum cleaner is given by $D(p)=\frac{-p^{2}}{100}+500$, where $p$ is the price in dollars. If the price, in terms of the cost $c$, is expressed as $p(c)=2 c-10$, find the demand in terms of the cost.

Consider the following table of values of the functions $f$ and $g$ and their derivatives at various points.

$$\begin{array}{|l|c|c|c|c|}\hline {x} & {1} & {2} & {3} & {4} \\ \hline f(x) & {2} & {4} & {1} & {3} \\ \hline {f^{\prime}(x)} & {-6} & {-7} & {-8} & {-9} \\\hline {g(x)} & {2} & {3} & {4} & {1} \\ \hline {g^{\prime}(x)} & {2 / 7} & {3 / 7} & {4 / 7} & {5 / 7}\\\hline\end{array}$$

Find the following using the table above. a. $D_{x}(g[f(x)])$ at $x=1 \quad$ b. $D_{x}(g[f(x)]) \quad$ at $x=2$

Find and solve the model for drug injection into the bloodstream if, beginning at t=0, a constant amount A g/min is injected and the drug is simultaneously removed at a rate proportional to the amount of the drug present at time t.

Some psychologists contend that the number of facts of a certain type that are remembered after $t$ hours is given by $f(t)=\frac{90 t}{99 t-90}.$ Find the rate at which the number of facts remembered is changing after the following numbers of hours.

$$a. 1 \qquad b. 10$$

Assume that a demand equation is given by $q=5000-100 p$ . Find the marginal revenue for the following production levels (values of $q$ ). (Hint: Solve the demand equation for $p$ and use $R(q)=q p . )$

$$a. 1000 \text{ units } \quad b. 2500 \text{ units } \quad c. 3000 \text{ units }$$

The total cost (in hundreds of dollars) to produce x units of perfume is C(x)=$\frac{3 x+2}{x+4}$
Find the average cost for each production level.
a) 10 units
b) 20 units
c) x units
d) Find the marginal average cost function.

A plucked violin string carries a traveling wave given by the equation

$$f ( x , t ) = a \sin \left[ b ( x - c t ) + \phi _ { \mathrm { i } } \right]$$

, with a = 0.00580 m, $b = 33.05 \mathrm { m } ^ { - 1 }$, and c = 245 m/s. (a) If f(0, 0) = 0, what is $\phi _ { \mathrm { i } }$? (b) Determine the simple harmonic period of a point on the string.

The formula for the growth rate of a population in the presence of a quantity $x$ of food was given as $f(x)=\frac{K x}{A+x}$ This was referred to as Michaelis-Menten kinetics. a. Find the rate of change of the growth rate with respect to the amount of food. b. The quantity $A$ in the formula for $f(x)$ represents the quantity of food for which the growth rate is half of its maximum. Using your answer from part a, find the rate of change of the growth rate when $x=A$ .

Economists have considered the output y of a manufacturing process as a function of the size of the labor force n using the function

$$y=k n^p$$

where 0<p<1. The marginal product of labor, defined as dy/dn, measures the rate that output increases with the size of the labor force, and is a measure of labor productivity.

(a) Show that

$$\frac{d y}{d n}=\frac{k p}{n^{1-p}} .$$

(b) How can you tell from your answer to part (a) that as the size of the labor force increases, the marginal product of labor gets smaller? This is a phenomenon known as the law of diminishing returns, discussed more in the next chapter.

Explain how the joints between vertebrae permit movement.

The brain mass of a human fetus during the last trimester can be accurately estimated from the circumference of the head by $m(c)=\frac{c^{3}}{100}-\frac{1500}{c}$ where $m(c)$ is the mass of the brain (in grams) and $c$ is the circumference (in centimeters) of the head. Source: Early Human Development. a. Estimate the brain mass of a fetus that has a head circumference of $30$ cm. b. Find the rate of change of the brain mass for a fetus that has a head circumference of $30$ cm and interpret your results.

In each part, determine whether the given vector p(t) in P_2 belongs to span $\left \{ p_{1}(t), p_{2}(t), p_{3}(t) \right \}$, where $p_{1}(t) = t^2 + 2t +1, p_{2}(t) = t^2 + 3, p_{3}(t)= t -1$ (a) p(t) = t^2 + t + 2 (b) p(t) = 2t^2 + 2t + 3 (c) p(t) = -t^2 + t - 4 (d) p(t) = -2t^2 + 3t + 1