3.4 Derivatives as rates of change  (Page 6/15)

A small town in Ohio commissioned an actuarial firm to conduct a study that modeled the rate of change of the town’s population. The study found that the town’s population (measured in thousands of people) can be modeled by the function $P\left(t\right)=-\frac{1}{3}{t}^3+64t+3000,$ where $t$ is measured in years.

1. Find the rate of change function $P^{\prime }\left(t\right)$ of the population function.
2. Find $P^{\prime }\left(1\right),P^{\prime }\left(2\right),P^{\prime }\left(3\right),$ and $P^{\prime }\left(4\right).$ Interpret what the results mean for the town.
3. Find $P\text{″}\left(1\right),P\text{″}\left(2\right),P\text{″}\left(3\right),$ and $P\text{″}\left(4\right).$ Interpret what the results mean for the town’s population.

[T] A culture of bacteria grows in number according to the function $N\left(t\right)=3000\left(1+\frac{4t}{t^2+100}\right),$ where $t$ is measured in hours.

1. Find the rate of change of the number of bacteria.
2. Find $N^{\prime }\left(0\right),N^{\prime }\left(10\right),N^{\prime }\left(20\right),$ and $N^{\prime }\left(30\right).$
3. Interpret the results in (b).
4. Find $N\text{″}\left(0\right),N\text{″}\left(10\right),N\text{″}\left(20\right),$ and $N\text{″}\left(30\right).$ Interpret what the answers imply about the bacteria population growth.

The centripetal force of an object of mass $m$ is given by $F\left(r\right)=\frac{m{v}^2}{r},$ where $v$ is the speed of rotation and $r$ is the distance from the center of rotation.

1. Find the rate of change of centripetal force with respect to the distance from the center of rotation.
2. Find the rate of change of centripetal force of an object with mass 1000 kilograms, velocity of 13.89 m/s, and a distance from the center of rotation of 200 meters.

[T]

1. Using a calculator or a computer program, find the best-fit linear function to measure the population.
2. Find the derivative of the equation in a. and explain its physical meaning.
3. Find the second derivative of the equation and explain its physical meaning.

[T]

1. Using a calculator or a computer program, find the best-fit quadratic curve through the data.
2. Find the derivative of the equation and explain its physical meaning.
3. Find the second derivative of the equation and explain its physical meaning.

[T]

1. Using a calculator or computer program, find the best-fit quadratic curve to the data.
2. Find the derivative of the position function and explain its physical meaning.
3. Find the second derivative of the position function and explain its physical meaning.