Chapter 9.5, Problem 13E

(a)

To determine

Approximately, how many people live within the ring about the city whose inner circle is at $t$ miles and outer circle is at $t+\Delta t$ miles, where the population density function for a city is $40e^{-0.5t}$ thousand people per square mile.

(b)

To determine

What does the expression $\frac{P\left(t+\Delta t\right)-P\left(t\right)}{\Delta t}$ approach as $\Delta t$ tends to zero, where the population density function for a city is $40e^{-0.5t}$ thousand people per square mile.

(c)

To determine

What does the quantity $P\left(5+\Delta 5\right)-P\left(5\right)$ represent, where the population density

function for a city is $40e^{-0.5t}$ thousand people per square mile.

(d)

To determine

The formula for $\frac{P\left(t+\Delta t\right)-P\left(t\right)}{\Delta t}$ and also obtain an approximate formula for the derivative $P\text{'}\left(t\right)$

(e)

To determine

The formula involving a definite integral, for the number of people who live in

the city between $a$ miles and $b$ miles of the city center, for two positive numbers $a$ and $b$.