Suppose you own a parcel of land whose value $t$ years from now will be $V(t)=8,000 e^{\sqrt{ }}$ dollars. If the prevailing interest rate remains constant at $6 \%$ per year compounded

Find the largest and smallest values of the given function over the prescribed closed, bounded interval.

• $g(x)=\frac{e^x}{2 x+1} \quad$ for $0 \leq x \leq 1$

A company organizes a training program in which it is determined that after $t$ weeks. the average trainee produces

$$P(t)=50\left(1-e^{-0.15 t}\right) \text { units }$$

while a typical new worker without special training produces

$$W(t)=\sqrt{150 t} \text { units }$$

a. How many units does the average trainee produce during the third week of the training period?

b. Explain how the function $F(t)=P(t)-W(t)$ can be used to evaluate the effectiveness of the training program. Is the program effective if it lasts just $5$ weeks? What if it lasts at least $7$ weeks? Explain your reasoning.

It is known that fluorocarbons have the effect of depleting ozone in the upper atmosphere. Suppose it is found that the amount of original ozone $Q_0$ that remains after $t$ years is given by

$$Q=Q_0 e^{-0.0015 t}$$

a. At what percentage rate is the ozone level decreasing at time $t$ ?

b. How many years will it take for $10 \%$ of the ozone to be depleted? At what percentage rate is the ozone level decreasing at this time?

Find the largest and smallest values of the given function over the prescribed closed, bounded interval.

• $f(x)=(3 x-1) e^{-x}$ for $0 \leq x \leq 2$

Find $\ln \frac{1}{\sqrt{a b^3}}$ if $\ln a=2$ and $\ln b=3$.

A manufacturer estimates that when $x$ units of a particular commodity are produced, the market price $p$ (dollars per unit) is given by the demand function

a. What market price corresponds to the production of $x=100$ units?

b. How much revenue is obtained when 100 units of the commodity are produced?

c. How much more (or less) revenue is obtained when $x=100$ units are produced than when $x=50$ are produced?

What is the present value of $\$ 10,000$ over a 5 -year period of time if interest is compounded continuously at an annual rate of $7 \%$ ? What is the present value of $\$ 20,000$ under the same conditions?

If $\log _3(x-5)=2$, what is $\ln x$ ?

An epidemic spreads through a community so that $t$ weeks after its outbreak, the number of people who have been infected is given by a function of the form $f(t)=\frac{B}{1+C e^{-k t}}$, where $B$ is the number of residents in the community who are susceptible to the disease. If $\frac{1}{5}$ of the susceptible residents were infected initially and $\frac{1}{2}$ had been infected by the end of the fourth week, what fraction of the susceptible residents will have been infected by the end of the eighth week?

Find the largest and smallest values of the given function over the prescribed closed, bounded interval.

• $F(x)=e^{x^2-2 x}$ for $0 \leq x \leq 2$

Differentiate the given function.

$$f(x)=\frac{e^x}{x}$$

Sketch the curves $y=\left(\frac{1}{3}\right)^x$ and $y=\left(\frac{1}{4}\right)^x$ on the same set of axes.

Evaluate the given expression using properties of the natural logarithm.

$$\ln \sqrt{e}$$