A bacterial colony is estimated to have a population of $P(t)=\frac{24 t+10}{t^{2}+1}$ million t hours after the introduction of a toxin. a. At what rate is the population changing 1 hour after the toxin is introduced (t = 1)? Is the population increasing or decreasing at this time? b. At what time does the population begin to decline?

Consider $f(x)=\frac{x}{1+x^{2}}$. Find the position and nature of all turning points of y = f(x).

A company manufactures a "thin" DVD burner kit that can be plugged into personal computers. The marketing manager determines that t weeks after an advertising campaign begins, P(t) percent of the potential market is aware of the burners, where $P(t)=100\left(\frac{t^{2}+5 t+5}{t^{2}+10 t+30}\right).$ a. At what rate is the market percentage P(t) changing with respect to time after 5 weeks? Is the percentage increasing or decreasing at this time? b. What happens to the percentage P(t) in the long run, that is, as $t \rightarrow +\infty?$ What happens to the rate of change of P(t) as $t \rightarrow +\infty?$

The manager of the Many Facets jewelry store models total sales by the function $S(t)=\frac{2,000 t}{4+0.3 t}$ where t is the time (years) since the year 2010 and S is measured in thousands of dollars. a. At what rate were sales changing in the year 2012? b. What happens to sales in the long run (that is, as $t \rightarrow +\infty )?$

An ornithologist determines that the body temperature of a certain bird species fluctuates over roughly a 17-hour period according to the cubic formula $T(t)=-68.07 t^{3}+30.98 t^{2}+12.52 t+37.1$ for $0 \leq t \leq 0.713,$ where T is the temperature in degrees Celsius measured t days from the beginning of a period. a. Compute and interpret the derivative T'(t). b. At what rate is the temperature changing at the beginning of the period (t = 0) and at the and of the period (t = 0.713)? Is the temperature increasing or decreasing at each of these times? c. At what time is the temperature not changing (neither increasing nor decreasing)? What is the bird's temperature at this time? Interpret your result.

Bea Johnson, the owner of the Bea Nice boutique, estimates that when a particular kind of perfume is priced at p dollars per bottle, she will sell $B(p)=\frac{500}{p+3} \quad p \geq 5$ bottles per month at a total cost of $C(p)=0.2 p^{2}+3 p+200$ dollars. a. Express Bea's profit P(p) as a function of the price p per bottle. b. At what rate is the profit changing with respect to p when the price is $12 per bottle? Is profit increasing or decreasing at that price?

The manager of a company that produces graphing calculators determines that when x thousand calculators are produced, they will all be sold when the price is $p(x)=\frac{1,000}{0.3 x^{2}+8}$ dollars per calculator. a. At what rate is demand p(x) changing with respect to the level of production x when 3,000 (x = 3) calculators are produced? b. The revenue derived from the sale of x thousand calculators is R(x) = xp(x) thousand dollars. At what rate is revenue changing when 3,000 calculators are produced? Is revenue increasing or decreasing at this level of production?

An electron starts from rest in a vacuum, in a region of strong electric field. The electron moves through a potential difference of 35 V. What would happen if the particle were a proton?

Everett has just received $500 as a birthday gift and has decided to spend it on DVDs and video games. He has determined that the utility (satisfaction) derived from the purchase of x DVDs and y video games is

$$U(x, y)=\ln \left(x^{2} \sqrt{y}\right)$$

If each DVD costs $20 and each video game costs$50, how many DVDs and video games should he purchase to maximize utility?

It is projected that x months from now, the population of a certain town will be

$$P(x)=2 x+4 x^{3 / 2}+5,000$$

(a.) At what rate will the population be changing with respect to time 9 months from now? (b.) At what percentage rate will the population be changing with respect to time 9 months from now?

Which security policies and procedures would you use as a business owner?

The population $P(t)$ (thousands) of a bacterial colony $t$ hours after the introduction of a toxin is changing at the rate $P^{\prime}(t)=(1-0.5 t) e^{0.5 t}$ thousand bacteria per hour By how much does the population change during the fourth hour?

A bacterial colony is estimated to have a population of

$$P(t)=\frac{24 t+10}{t^2+1}$$

million $t$ hours after the introduction of a toxin.

a. At what rate is the population changing 1 hour after the toxin is introduced $(t=1)$ ? Is the population increasing or decreasing at this time?

b. At what time does the population begin to decline?

An ornithologist determines that the body temperature of a certain species of bird fluctuates over roughly a 17-hour period according to the cubic formula $T(t)=-68.07 t^3+30.98 t^2+12.52 t+37.1$ for $0 \leq t \leq 0.713$, where $T$ is the temperature in degrees Celsius measured $t$ days from the beginning of a period.

a. Compute and interpret the derivative $T^{\prime}(t)$.

b. At what rate is the temperature changing at the beginning of the period $(t=0)$ and at the end of the period $(t=0.713)$ ? Is the temperature increasing or decreasing at each of these times?

c. At what time is the temperature not changing (neither increasing nor decreasing)? What is the bird's temperature at this time? Interpret your result.