Average and Instantaneous Rate of Change

Instantaneous Rate Of Change: We see changes around us everywhere. When we project a ball upwards, its position changes with respect to time, and its velocity changes as its position changes. The height of a person changes with time. The prices of stocks and options change with time. The equilibrium price of good changes with respect to demand and supply. The power radiated by a black body changes as its temperature changes. The surface area of a sphere changes as its radius changes. This list never ends. It is amazing to measure and study these changes.

Imagine that you drive to a grocery store 10 miles away from your house, and it takes you 30 minutes to get there. That means that you traveled 10 miles in 1/2 hour, at an average speed of 20 miles per hour. (10 miles divided by 1/2 hour = 20 miles per hour). The speed of your car is a great example of a rate of change.

Average and Instantaneous Rate of Change

A rate of change tells you how quickly something is changing, such as the location of your car as you drive. You can also measure how quickly your hair grows, how much money your business makes each month, or how much water flows over a dam. All of these, and many more, can be represented by calculating the average rate of change of a quantity over a certain amount of time.

One easy way to calculate a rate of change is to make a graph of the quantity that is changing versus time. Then you can calculate the rate of change by finding the slope of the graph, like this one. The slope is found by dividing how much the y values change by how much the x values change. Let’s look at a graph of position versus time and use that to determine the rate of change of position, more commonly known as speed.

These changes depend on many factors; for example, the power radiated by a black body depends on its surface area as well as temperature. We shall be looking at cases where only one factor is varying and all others are fixed. Then we can model our system as $y = f (x),$ where $y$ changes with regard to $x$ .

Instantaneous Rate Of Change Calculator

So, we saw that you could calculate the average rate of change by calculating the slope of a line, but does that work for instantaneous rates of change as well? In fact, it does, although you have to think about slope a little differently than you may have before.

If you have a graph of your position vs. time that is NOT a straight line and you want to calculate your instantaneous speed, you can draw a line, known as a tangent line, that only touches the graph at one point. The slope of this tangent line will give you the instantaneous rate of change at exactly that point.

As you can see from the calculation on this graph, v equals 20 meters divided by 5 seconds minus 1.5 seconds, meaning 3.5 seconds, which equals 5.7 meters per second. How does that compare to the average rate of change? To determine your average speed over the whole trip, calculate the slope of a line drawn from the first point on the graph to the last point.

Instantaneous Rate Of Change Formula

A car is travelling on a straight road parallel to the $x$ -axis. At $t = 0$ seconds, the car is at $x = 2$ meters; at $t = 6$ seconds, the car is at $x = 14$ meters. Find the average rate of change of the $x$ -coordinate of the car with respect to time.

Using the formula, we get

$\text{Rate} = \dfrac{\Delta x}{\Delta t} = \dfrac{14 – 2}{6 – 0} = 2 \text{ m/s}.\ _\square$

How To Find Instantaneous Rate Of Change

Let’s go back a moment and think about that grocery store trip again. We calculated that your average speed for the entire trip was 20 miles per hour, but does that mean that you were traveling at exactly 20 miles per hour for the entire trip? What about when you were stopped at a red light or were stuck in traffic that wasn’t moving? During those times you weren’t moving at all, so your speed was zero.

When you measure a rate of change at a specific instant in time, this is called an instantaneous rate of change. An average rate of change tells you the average rate at which something was changing over a longer time period. While you were on your way to the grocery store, your speed was constantly changing. Sometimes you were moving faster than 20 miles per hour and sometimes slower. At each instant in time, your instantaneous rate of change would correspond to your speed at that exact moment.

Instantaneous Rate Of Change Calculus

First, both of these problems will lead us into the study of limits, which is the topic of this chapter after all. Looking at these problems here will allow us to start to understand just what a limit is and what it can tell us about a function.

Secondly, the rate of change problem that we’re going to be looking at is one of the most important concepts that we’ll encounter in the second chapter of this course. In fact, it’s probably one of the most important concepts that we’ll encounter in the whole course. So, looking at it now will get us to start thinking about it from the very beginning.

Tangent Lines

The first problem that we’re going to take a look at is the tangent line problem. Before getting into this problem it would probably be best to define a tangent line.

A tangent line to the function $f (x)$ at the point $x = a$ is a line that just touches the graph of the function at the point in question and is “parallel” (in some way) to the graph at that point. Take a look at the graph below.

$A graph of some (unknown) function as well as a line. Indicated on the graph is a point where the line forms a tangent line to the graph of the function (i.e. is parallel to the graph of the function) and a point where the line does not form a tangent line to the graph of the function (i.e. is not parallel to the graph of the function).$

In this graph the line is a tangent line at the indicated point because it just touches the graph at that point and is also “parallel” to the graph at that point. Likewise, at the second point shown, the line does just touch the graph at that point, but it is not “parallel” to the graph at that point and so it’s not a tangent line to the graph at that point.

At the second point shown (the point where the line isn’t a tangent line) we will sometimes call the line a secant line.

We’ve used the word parallel a couple of times now and we should probably be a little careful with it. In general, we will think of a line and a graph as being parallel at a point if they are both moving in the same direction at that point. So, in the first point above the graph and the line are moving in the same direction and so we will say they are parallel at that point. At the second point, on the other hand, the line and the graph are not moving in the same direction so they aren’t parallel at that point.

Okay, now that we’ve gotten the definition of a tangent line out of the way let’s move on to the tangent line problem. That’s probably best done with an example.