### Subsection2.3.3Exercises

###### 1

Suppose $h(t)$ gives the height at time $t$ of the water at a dam, where the units of $t$ are hours and the units of $h$ are meters.

1. What is the physical interpretation of the slope of the secant line through the points $(0,h(0))$ and $(24,h(24))\text{?}$
2. What is the physical interpretation of the slope of the tangent line to the curve $y=h(t)$ at the point $(0,h(0))\text{?}$
###### 2

Suppose $p(t)$ is a function that gives the profit generated by selling $t$ widgets. What is the practical interpretation of $p'(t)\text{?}$

###### 3

$T(d)$ gives the temperature of water at a particular location $d$ metres below the surface. What is the physical interpretation of $T'(d)\text{?}$ Would you expect the magnitude of $T'(d)$ to be larger when $d$ is near 0, or when $d$ is very large?

###### 4

$C(w)$ gives the calories in $w$ grams of a particular dish. What does $C'(w)$ describe?

###### 5

The velocity of a moving object at time $t$ is given by $v(t)\text{.}$ What is $v'(t)\text{?}$

###### 6

The function $T(j)$ gives the temperature in degrees Celsius of a cup of water after $j$ joules of heat have been added. What is $T'(j)\text{?}$

###### 7

A population of bacteria, left for a fixed amount of time at temperature $T\text{,}$ grows to $P(T)$ individuals. Interpret $P'(T)\text{.}$

###### 8

You hammer a small nail into a wooden wagon wheel. $R(t)$ gives the number of rotations the nail has undergone $t$ seconds after the wagon started to roll. Give an equation for how quickly the nail is rotating, measured in degrees per second.

###### 9

A population of bacteria, left for a fixed amount of time at temperature $T\text{,}$ grows to $P(T)$ individuals. There is one ideal temperature where the bacteria population grows largest, and the closer the sample is to that temperature, the larger the population is (unless the temperature is so extreme that it causes all the bacteria to die by freezing or boiling). How will $P'(T)$ tell you whether you are colder or hotter than the ideal temperature?