SOCIAL SCIENCE: Diffusion of Information by Mass Media In a city of a million people, election results broadcast over radio and television will reach $N(t)=1,000,000\left(1-e^{-0.3 t}\right)$ people within t hours. Find when the news will have reached 500,000 people.

CHANGE OF BASE FORMULA FOR

LOGARITHMS Derive the formula

$$\log _a x=\frac{\ln x}{\ln a} \quad(\text { for } a>0 \text { and } x>0 \text { ) }$$

which expresses logarithms to any base a in terms of natural logarithms, as follows:

a. Define $y=\log _a x$, so that x=ay, and take the natural logarithms of both sides of the last equation and obtain $\ln x=y \ln a$.

b. Solve the last equation for y to obtain $y=\frac{\ln x}{\ln a}$ and then use the original definition of y to obtain the stated change of base formula.

BIOMEDICAL: Drug Concentration If a drug is injected intramuscularly, the concentration of the drug in the bloodstream after t hours will be

$$A(t)=\frac{c}{b-a}\left(e^{-a t}-e^{-b t}\right)$$

If the constants are a=0.4, b=0.6, and c=0.1, find the time of maximum concentration.

BIOMEDICAL: Reynolds Number An important characteristic of blood flow is the Reynolds number. As the Reynolds number increases, blood flows less smoothly. For blood flowing through certain arteries, the Reynolds number is

$$R(r)=a \ln r-b r$$

where a and b are positive constants and r is the radius of the artery. Find the radius r that maximizes the Reynolds number R. (Your answer will involve the constants a and b.)

BIOMEDICAL: Ricker Recruitment The population dynamics of many fish (such as salmon) can be described by the Ricker curve $y=a x e^{-b x}$ for $x \geq 0$ where a>1 and b>0 are constants, x is the size of the parental stock, and y is the number of recruits (offspring). Determine the size of the parental stock that maximizes the number of recruits.

BIOMEDICAL: Population Growth The Gompertz growth curve models the size N(t) of a population at time $t \geq 0$ as $N(t)=K e^{-a e^{-t t}}$ where K and b are positive constants. Show that $\frac{d N}{d t}=b N \ln \left(\frac{K}{N}\right)$ and interpret this derivative to make statements about the population growth when NK.

Explain why it is obvious, without any calculation, that $\frac{d}{d x} \ln e^x=1$

Explain why it is obvious, without any calculation, that $\frac{d}{d x} e^{\ln x}=1$.

True or False: If a function involves natural logarithms, then its derivative will involve natural logarithms. Explain.

Choose the correct answer: $\frac{d}{d x} \ln 5=$

a. $\frac{5}{1}$

b. $\frac{1}{5}$

c. 0

Choose the correct answer: $\frac{d}{d x} \ln f(x)=$

a. $\frac{f(x)}{f^{\prime}(x)}$

b. $\frac{f^{\prime}(x)}{f(x)}$

c. $\frac{f^{\prime}(x)}{f^{\prime}(x)}$

Find the derivative of the function $f(x)=\ln (5 x)$

Evaluate the expression $e^{-0.09}$ using a calculator.

For each function: $f(t)=100 \sqrt[3]{t+2}, t=8$

a. Find the relative rate of change.

b. Evaluate the relative rate of change at the given value(s) of t.