Calculus I, Chapter 3: The Average Rate of Change and Limits
Previous Section: 3.1 The Average Rate of Change
Distinguishing Rates of Change
In the previous chapter we introduce the Average Rate of Change as a way to approximate a curved line as a linear one in order to determine how it changes over time. But, this is not helpful if we want meaningful and specific information. If we were to describe a functions change over a large interval we are forgetting about so much information that happens in the middle! Take the following two graphs as an example:
They both have an average rate of change of about (-2)/3 over the interval [0,2.983] while the function does dramatically different things in between.
Another issue arises when trying to find the rate of change of a function with an asymptote:
Calculations show that there is an average rate of change of 1 over the interval [2,4]. However, this sort of information is meaningless here. So, there should be some specific rules that need to be applied in order for this to work properly.
Modifying the Average Rate of Change Formula
We can begin by trying to approximate the curve on smaller intervals in order to get a more precise picture. The easiest way to do this is to think of our definition for rate of change. Recall:
average rate of change= (f(x_2 )-f(x_1))/(x_2-x_1 )
This can be rephrased so that the distance between x_1 and x_2 is ∆x. That is x+∆x=x_2 :
rate of change= (f(x+∆x)-f(x))/((x+∆x)-x)
If we attempt to take a rate of change over a very small interval, what we are saying is that the distance between the two points is approaching zero. It is hard to imagine this, so we can use this graph and the lines below to demonstrate the process:
Here is a visualization with all of the secant lines drawn on the same graph and zoomed in a bit:
In essence, we can define:
instantaneous rate of change at x= (f(x+∆x)-f(x))/∆x as ∆x approaches 0
Visually, we can see that the line that we are drawing to represent the slope is not intersecting the curve twice anymore, but instead just barely touches the curve at one point. When this happens, we call the line a tangent line. So, the instantaneous rate of change is represented as a line tangent to the curve at a point. It’s okay to think of it as at a single point because what we have done is defined it so that the distance between the points is practically 0; the points are basically the same single point!
Example 1: Find the Instantaneous Rate of Change of f(x)=x^2 at x=2.
Solution:
(f(x+∆x)-f(x))/∆x=((x+∆x)^2-x^2)/∆x
=((2+∆x)^2-2^2)/∆x
=((2+0)^2-2^2)/0 if ∆x=0
=(4-4)/0
=0/0
Uh oh! This is an indeterminate value and doesn’t have any real meaning here. This example doesn’t seem to have a solution yet.
Solving for the instantaneous rate of change leads to a major problem, though. Every time we plug in 0 for ∆x we end up with 0/0! For now, this is impossible to solve. The best we can do is approximate the instantaneous rate of change by using a chart and plugging in smaller and smaller values. It is important to plug in values on both sides of the x-value whose slope we are approximating. Let’s try the first example again.
Example 1, again: Find the Instantaneous Rate of Change of f(x)=x^2 at x=2.
Solution: We will solve this in parts. First by choosing some intervals around the value x=2 and secondly by evaluating an Average Rate of Change on those intervals.
For example, on the interval [1.9, 2]:
(2^2-〖1.9〗^2)/(2-1.9)=3.9
| Interval | Average Rate of Change |
| [1.9,2] | 3.9 |
| [1.99,2] | 3.99 |
| [1.9999,2] | 3.999 |
| [2,2] (at x=2) | Instantaneous Rate of Change at x=2 |
| [2,2.0001] | 4.001 |
| [2,2.01] | 4.01 |
| [2,2.1] | 4.1 |
As we can see, the chart demonstrates that the values of the chart are getting closer and closer to a specific number. That number is the instantaneous rate of change. So, the instantaneous rate of change of f(x)=x^2 at x=2 is 4.
Example 2: Find the Instantaneous Rate of Change of f(x)=2√(x+3) at x= -1.
Solution:
| Interval | Average Rate of Change |
| [-1.1,-1] | -0.716174 |
| [-1.01,-1] | -0.707993 |
| [-1.0001,-1] | -0.707116 |
| at x=-1 | Instantaneous Rate of Change at x=-1 |
| [-1,-0.9999] | -0.707098 |
| [-1,-0.99] | -0.706225 |
| [-1,-0.9] | -0.698482 |
The instantaneous rate of change can be estimated as -0.7071, which is very close to √2.
The only event in which an Instantaneous Rate of Change does not exist is one where this process doesn’t seem to show that the result approaches a specific value at the point of interest. Here is another example of a function with no solution for the instantaneous rate of change at a particular point.
Example 3: Find the Instantaneous Rate of Change of f(x)= cosx/x at x=0.
Solution:
| Interval | Average Rate of Change |
| [-0.1,0] | |
| [-0.01,0] | |
| [-0.0001,0] | |
| at x=0 | Instantaneous Rate of Change at x=0 |
| [0,0.0001] | |
| [0,0.01] | |
| [0,0.1] |
Review
The Instantaneous Rate of Change is the slope at a single point of a function.
The line representing the instantaneous rate of change touches the curve at a single point and is called a tangent line.
Evaluating the instantaneous rate of change results in 0/0, so it needs to be estimated using a chart and several average rates of changes in order to do so accurately
If the chart method does not demonstrate the result to be approaches a single answer, then the instantaneous rate of change does not exist
Equations
instantaneous rate of change at x= (f(x+∆x)-f(x))/∆x as ∆x approaches 0