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
where
is measured in years.
Find the rate of change function
of the population function.
Find
and
Interpret what the results mean for the town.
Find
and
Interpret what the results mean for the town’s population.
[T] A culture of bacteria grows in number according to the function
where
is measured in hours.
Find the rate of change of the number of bacteria.
Find
and
Interpret the results in (b).
Find
and
Interpret what the answers imply about the bacteria population growth.
a.
b.
c. The bacteria population increases from time 0 to 10 hours; afterwards, the bacteria population decreases. d.
The rate at which the bacteria is increasing is decreasing during the first 10 hours. Afterwards, the bacteria population is decreasing at a decreasing rate.
The centripetal force of an object of mass
is given by
where
is the speed of rotation and
is the distance from the center of rotation.
Find the rate of change of centripetal force with respect to the distance from the center of rotation.
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.
For the following exercises, consider an astronaut on a large planet in another galaxy. To learn more about the composition of this planet, the astronaut drops an electronic sensor into a deep trench. The sensor transmits its vertical position every second in relation to the astronaut’s position. The summary of the falling sensor data is displayed in the following table.
Time after dropping (s)
Position (m)
0
0
1
−1
2
−2
3
−5
4
−7
5
−14
[T]
Using a calculator or computer program, find the best-fit quadratic curve to the data.
Find the derivative of the position function and explain its physical meaning.
Find the second derivative of the position function and explain its physical meaning.
a.
b.
This is the velocity of the sensor. c.
This is the acceleration of the sensor; it is a constant acceleration downward.