Differentiation V: Derivatives and Rates of Change
Primary tabs
SAMPLE LESSON
Sign in for easy to read lessons narrowly focused on a specific subject. All lessons are linked to each other allowing you to easily see the connections between different Calculus concepts. Our unique Calcmap will give you a bird's-eye view of all our lessons and help you see how important concepts are related to each other. All of the formulas are available to our subscribers, but here is a sample of the lesson you will see when you log in:
A rate of change is a number that expresses how much one variable changes in relation to another. The most common example of this is speed. If you are traveling down the highway at a speed of 75 miles per hour then your position on the highway will change by 75 miles for every hour you drive. So your speed is a rate of change that measures your position on the highway in relation to how long you have traveled.
Now let's do a thought experiment. Suppose that you get into a car and begin driving down the highway. Every hour you write down on a piece of paper your distance from home and the amount of time you have been traveling. (We will of course measure distance in miles and the time in hours.) Here is our data:
| Hour | $0$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|---|
| Distance | $0$ | $55$ | $130$ | $180$ | $250$ | $300$ |
Using the data above we can then calculate the speed by taking the distance traveled and divide it by the amount of time taken. Now let's place our speed in the table:
| Hour | $0$ | $1$ | $2$ | $3$ | $4$ | $5$ | |||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Distance | $0$ | $55$ | $130$ | $180$ | $250$ | $300$ | |||||
| Speed | <Sign in to see all the formulas> | <Sign in to see all the formulas> | <Sign in to see all the formulas> | <Sign in to see all the formulas> | <Sign in to see all the formulas> |
It is important to note that the speed does not occur at one point, but is the speed driven between two points of time and across a certain distance. That is why we have written it between the time and distances they correspond to. We can also make a graph where we plot the time on the x-axis and distance on the y-axis. We have included the speed where appropriate. Notice that the higher the speed the steeper the slope of the graph.
Now we can put this into some math language by denoting a specific time as $t_i$ and a specific distance as $D_i$. The speed between times $t_i$ and <Sign in to see all the formulas> is denoted by <Sign in to see all the formulas> and is
<Sign in to see all the formulas>
So now our table looks like:
| Hour | $t_0=0$ | $t_1=1$ | $t_2=2$ | $t_3=3$ | $t_4=4$ | $t_5=5$ | |||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Distance | $D_0=0$ | <Sign in to see all the formulas> | <Sign in to see all the formulas> | <Sign in to see all the formulas> | <Sign in to see all the formulas> | <Sign in to see all the formulas> | |||||
| Speed | <Sign in to see all the formulas> | <Sign in to see all the formulas> | <Sign in to see all the formulas> | <Sign in to see all the formulas> | <Sign in to see all the formulas> |
Now let's alter our thought experiment. Instead of writing down our distance every hour let's write it down every 15 minutes. Our table is going to be much too large to write every data point down, but we can write out the first few:
We can also graph this out to see all of the data pictorially. So let's gain some intuition about what is going on here. At the beginning we are only driving 20mph. This is probably because we are driving through traffic in the city to get to the highway. Once we get to the highway we speed up. Somewhere in the middle of our trip we have a fifteen minute period of time where our speed is zero. This was a rest stop to get gas and something to eat. Shortly after this we have a fifteen minute period where we are only going 28mph. This is because we hit some construction and had to slow down. Later on we found that we lost some time because of the construction and decided to go 104mph! Then at the end we slow down again. This is because we got off the highway and back into the city where we had to slow down before getting to our final destination. Again, it is important to recognize that the slope of the graph gets steeper as we speed up.
As we record our distance and time at smaller and smaller time steps we get a more complete picture of our speed during the trip. Taking smaller and smaller time steps means that <Sign in to see all the formulas> approaches zero. (<Sign in to see all the formulas> was hourly and then it was every fifteen minutes. Now think of <Sign in to see all the formulas> as every minute, then every second, and then every millisecond.) But how can we explain this process of taking smaller and smaller time steps? We can do this with function limits. As we take more and more data, we don't just create a table of data, but we get distance as a function of time. We get the function $D(t)$. Now if we want to find the speed at one point in time we take the limit
<Sign in to see all the formulas>
But this is just the derivative of $D(t)$ with respect to $t$! In other words,
<Sign in to see all the formulas>
So let's be very clear that we currently have two definitions of speed
<Sign in to see all the formulas>
The first is the average speed between times $t_i$ and <Sign in to see all the formulas> and the second is the instantaneous speed. We call them this because if we take the average of the instantaneous speeds, $S(t)$, between $t_i$ and <Sign in to see all the formulas>, then we get <Sign in to see all the formulas>. (There is actually more on this subject that requires a knowledge of integration. You may find more here if you are interested.) Here is a more formal summary of what we have learned:
The average rate of change of a function $f(x)$ between two points $x_1$ and $x_2$ is<Sign in to see all the formulas>where <Sign in to see all the formulas>. The function's instantaneous rate of change with respect to $x$ is its derivative $f'(x)$. <Sign in to see all the formulas>In this lesson $f$ would be the distance and $x$ would be the time. |
Finally, we can use the Leibniz Notation to put this into a format that is easily remembered:
<Sign in to see all the formulas> |
Here are more examples:
- Growth rate of Bob's Automotive repair shops.
- ACME Soap Company Sales Revenue Growth
- Suppose that your distance from home on a Sunday afternoon drive was given by the formula <Sign in to see all the formulas> where $D$ is the distance in miles and $t$ is the number of hours driven. Determine your speed during your two hour trip.
- The rate of change of the volume of a sphere with respect to its radius.
