- How can we rewrite the average rate of change of a function in terms of the horizontal distance, h, between the points? Why would we want to do that?
- What are some algebra techniques that allow us to simplify the average rate of change for an arbitrary h?
- What does it tell us when we put in small values for h?
Introduction
We have discussed a secant line to the graph of a function from to , and the fact that the slope, , of this line is the average rate of change of the function on the interval , . Furthermore, recall that we can let , so that the slope expression becomes where is the horizontal distance between the points.
One of the main objectives of Calculus is to understand instantaneous rates of change, as opposed to average rates of change. Namely, what is the behavior of the expression when gets very small? Geometrically, making become very small is making the secant line through and approach a certain line — the tangent line to the graph of at the point . This is demonstrated in the figure below, where the secant line is blue, and the line tangent to the graph at is in red.
Follow the link below to an example Desmos graph where you can see the effect of changing the value of on the secant line in real-time.
The slope of such a tangent line, when it exists, is called the derivative of at . Here, we’ll discuss difference quotients and several examples, to prepare you to learn those things in more detail in a future Calculus class.
Definitions and examples
- a.
- , .
Let’s evaluate it directly: We thus have an equation for the slope of the secant line from to : Recall from an example in the previous section, that we calculated the slope of the secant line from to . Letting , we see that this gives the same answer for the slope of that secant line.
Furthermore, if we now let , this expression, . This means that as gets smaller and smaller, the difference quotient gets closer and closer to 4.
- b.
- .Again, we evaluate directly: Recognizing that we cannot further simplify this expression in its current form, we replace using the angle sum formula: This gives a less easy to visualize equation for the slope of the secant line from to , for : However, consider , so that we are looking at the secant line from to . We then see that as before.
Furthermore, see what happens as we let , in essence, what happens as becomes smaller. Consider , then we have
Note that this is greater than 0. Think about the graph of . It is increasing on the interval .Follow the Desmos link to explore more initial values of and see what happens as you adjust smaller and smaller to zero.
- c.
- .Evaluate directly: Observe that the equation for the slope of the secant line from to is simply
Recall from the example in the previous section, that the equation of the secant line from to was simply . Why was this?
Now, this tells us that regardless of the points we choose, the secant line between them will have the same slope and equation as the line itself.