§7.1 An introduction to differential equations

Introduction

What is a differential equation?

Differential equations in the world around us

The position of a moving object is given by the function $s(t)$, where $s$ is measured in feet and $t$ in seconds. We determine that the object's velocity is $v(t) = 4t + 1$ feet per second.

How much does the position change over the time interval $[0,4]$?

The population of a town grows continuously at an annual rate of 1.25%.

A radioactive sample loses mass at a rate of 5.6% of its mass every day.

You have a bank account that continuously earns 4% interest every year. At the same time, you withdraw money continually from the account at the rate of ＄1000 per year.

A cup of hot chocolate is sitting in a 70 $^\circ$ room. The temperature of the hot chocolate cools continuously by 10% of the difference between the hot chocolate's temperature and the room temperature every minute.

A can of cold soda is sitting in a 70 $^\circ$ room. The temperature of the soda warms continuously at the rate of 10% of the difference between the soda's temperature and the room's temperature every minute.

Begin with the skydiver's velocity $v$ given by the graph at left below, which shows how $v$ changes as $t$ changes. Use this graph to find the rate of change $dv/dt$ at the points where the velocity is $v=0.5, 1.5, 2.0$, and $2.5$. Note particularly that several key slopes are provided for you on the given graph.

Then, on the axes provided, plot the values of the derivative $dv/dt$ as a function of velocity. Think carefully about this: we are focusing on how we can connect the velocity $v$ to the resulting value of $dv/dt$. For example, in the velocity graph at left, we observe from one of the points that when $v = 2$, $dv/dt = 0.5$.

Next we're going to consider a falling meteorite's velocity.

While the skydiver fell faster and faster, the meteorite falls slower and slower through the atmosphere, as shown in the graph at right. Note the scale on the vertical axis: since the velocity is always greater than $v = 3$, we only show $v$ values from $3$ to $6$. Reasoning similarly to part (a), use the given graph of the meteorite's velocity to find the rate of change $dv/dt$ at the points where the velocity is $v=3.5,4.0,4.5$, and $5.5$. Plot the appropriate resulting points on the same axes used in part (a).

You should find that all of the points you plotted on the axes in (a) lie on a line. Remember that these points show how $dv/dt$ depends on $v$. Write the equation of this line, being careful to use proper notation for the quantities on the horizontal and vertical axes.

The relationship you just found in (c) is a differential equation. Write a complete sentence that explains its meaning.

Use the differential equation you found in (c) to determine the values of the velocity for which the velocity increases. Write a sentence to explain your thinking.

Similarly, use the differential equation to determine the values of the velocity for which the velocity decreases. Again, explain your thinking.

Finally, determine the value(s) of the velocity for which the velocity remains constant. What do these values mean in the overall context of this activity?