The manager of the Footloose sandal company determines that $t$ months after initiating an advertising campaign, $S(t)$ hundred pairs of sandals will be sold, where

$$S(t)=\frac{3}{t+2}-\frac{12}{(t+2)^2}+5$$

a. Find $S^{\prime}(t)$ and $S^{\prime \prime}(t)$.

b. At what time will sales be maximized? What is the maximum level of sales?

c. The manager plans to terminate the advertising campaign when the sales rate is minimized. When does this occur? What are the sales level and sales rate at this time?

A company estimates that if x thousand dollars are spent on marketing a certain product, then S(x) units of the product will be sold each month, where $S(x)=-x^{3}+33 x^{2}+60 x+1,000.$ a. How many units will be sold if no money is spent on marketing? b. Sketch the graph of S(x). For what value of x does the graph have an inflection point? What is the significance of this marketing expenditure?

A company estimates that when x thousand dollars are spent on the marketing of a certain product, Q(x) units of the product will be sold, where $Q(x)=-4 x^{3}+252 x^{2}-3,200 x+17,000$ for $10 \leq x \leq 40.$ Sketch the graph of Q(x). Where does the graph have an inflection point? What is the significance of the marketing expenditure that corresponds to this point?

The profit obtained from producing x thousand units of a particular commodity each year is P(x) dollars, where $P(x)=-x^{9 / 2}+90 x^{7 / 2}-5,000.$ a. Find the marginal profit P'(x), and determine all values of x such that P'(x) = 0. b. Sketch the graph of marginal profit along with the graph of P(x) on the same coordinate plane. c. Find P''(x), and determine all values of x such that P''(x) = 0. How are these levels of production related to the graph of marginal profit?

Buses arrive at a station every 12 minutes, so if you arrive at a random time, your wait is uniformly distributed on the interval [0, 12]. Find: a. E(X), b. Var(X), c. $\sigma(X)$, d. $P(X \leq 3)$

A manufacturer can produce digital recorders at a cost of $50 apiece. It is estimated that if the recorders are sold for p dollars a piece, consumers will buy q = 120 - p recorders each month. a. Express the manufacturer's profit P as a function of q. b. What is the average rate of change in profit obtained as the level of production increases from q = 0 to q = 20? c. At what rate is profit changing when q = 20 recorders are produced? Is the profit increasing or decreasing at this level of production?

An efficiency study of the morning shift at a certain factory indicates that an average worker who is on the job at $8.00$ A.M. will have assembled $f(x)=-x^3+6 x^2+15 x$ units $x$ hours later. The study indicates further that after a 15-minute coffee break the worker can assemble $g(x)=-\frac{1}{3} x^3+x^2+23 x$ units in $x$ hours.

Determine the time between $8.00$ A.M. and noon at which a 15 -minute coffee break should be scheduled so that the worker will assemble the maximum number of units by lunchtime at $12: 15$ P.M. [Hint: If the coffee break begins $x$ hours after 8:00 A.M., $4-x$ hours will remain after the break.]

A company estimates that if x thousand dollars are spent on the marketing of a certain product, then Q(x) thousand units of the product will be sold, where $Q(x)=\frac{7 x}{27+x^{2}}.$ a. Sketch the graph of the sales function Q(x). b. For what marketing expenditure x are sales maximized? What is the maximum sales level? c. For what value of x is the sales rate minimized?

In this exercise, solve the differential equation.

$$\dfrac{dy}{dx}=\left(1+e^x\right)^2$$

An efficiency study of the morning shift at a certain factory indicates that an average worker who arrives on the job at 8:00 A.M. will have assembled

$$f(x)=-x^{3}+6 x+15 x^{2}$$

iPads x hours later $(0 \leq x \leq 8)$. (a.) How many units will such a worker have assembled by 10:00 A.M.? (b.) How many units will such a worker assemble between 9:00 A.M. and 10:00 A.M.?

An efficiency study of the morning shift at a certain factory indicates that an average worker arriving on the job at 8:00 A.M. will have assembled $f(x)=-x^{3}+6 x^{2}+15 x$ units x hours later. Approximately how many units will the worker assemble between 9:00 and 9:15 A.M.?

Use increments to make the required estimate.

An efficiency study of the morning shift at a certain factory indicates that an average worker arriving on the job at 8:00 A.M. will have assembled $f(x)=-x^3+6 x^2+15 x$ units $x$ hours later. Approximately how many units will the worker assemble between 9:00 and 9:15 A.M.?

An efficiency study of the morning shift (from 8:00 A.M. to 12:00 noon) at a certain factory indicates that an average worker who arrives on the job at 8:00 A.M. will have assembled $Q(t)=-t^3+6 t^2+15 t$ units $t$ hours later.

a. At what time during the morning is the worker performing most efficiently?

b. At what time during the morning is the worker performing least efficiently?

An efficiency study of the morning shift at a certain factory indicates that an average worker who arrives on the job at 8:00 A.M. will have assembled $f(x)=-x^{3}+6 x^{2}+15 x$ television sets x hours later. a. How many sets will such a worker have assembled by 10:00 A.M.? [Hint: At 10:00 A.M., x = 2.] b. How many sets will such a worker assemble between 9:00 and 10:00 A.M.?