According to the results* of Tucker and Schmidt-Koenig, the energy expended by a certain species of parakeet is given by

$$E(v)=\frac{1}{v}\left[0.074(v-35)^2+22\right]$$

where $v$ is the bird's velocity (in $\mathrm{km} / \mathrm{hr}$ ).

a. What velocity minimizes energy expenditure?

b. Read an article on how mathematical methods can be used to study animal behavior, and write a paragraph on whether you think such methods are valid. You may wish to begin with the reference cited in this exercise.

Use calculus to sketch the graph of the given function.

$$f(x)=3 x^5-5 x^3+4$$

A cruise line estimates that when each deluxe balcony stateroom on a particular cruise is priced at $p$ thousand dollars, then $q$ tickets for staterooms will be demanded by travelers, where $q=300-0.7 p^2$.

a. Find the elasticity of demand for the stateroom tickets.

b. When the price is $\$ 8,000(p=8)$ per stateroom, should the cruise line raise or lower the price in order to increase total revenue?

Find constants $A$ and $B$ so that the graph of the function

$$f(x)=\frac{A x-3}{5+B x}$$

will have $x=2$ as a vertical asymptote and $y=4$ as a horizontal asymptote. Once you find $A$ and $B$, sketch the graph of $f(x)$.

The first derivative $f^{\prime}(x)$ of a certain function $f(x)$ is given. In each case,

(a) Find intervals on which $f$ is increasing and decreasing.

(b) Find intervals on which the graph of $f$ is concave up and concave down.

(c) Find the $x$ coordinates of the relative extrema and inflection points of $f$.

(d) Sketch a possible graph for $f(x)$.

$$f^{\prime}(x)=x^2-4 x$$