While it can seem daunting, the difference quotient is a great tool to find the slope of the secant line of a curve. This lesson will break down the difference quotient into manageable steps so that you can become an expert at using this occasionally tricky formula.
When you first heard the term ‘difference quotient’, you may have drawn a blank. After all, it involves so many elements, like functions, and secants, and graphs, and even worse, a crazy formula that brings it all together. However, let me assure you, once you finish this lesson, you will definitely be an expert at using the difference quotient. Let’s start with the definition: The difference quotient is used to calculate the slope of the secant line between two points on the graph of a function, f.
Differential Quotient Graph
Just to review, a function is a line or curve that has only one y value for every x value. It’s like an input/output machine. For any number x that you plug into the function, you will get an output value for f (x).
In simple terms, the difference quotient helps us find the slope when we are working with a curve. In the case of a curve, we cannot use the traditional formula of:
which is why we must use the difference quotient formula.
In the formal definition of the difference quotient, you’ll note that the slope we are calculating is for the secant line. A secant line is just any line that passes between two points on a curve. We label these two points as x and (x +h) on our x-axis. Because we are working with a function, these points are labeled as f (x) and f (x + h) on our y-axis, respectively.
Now that we understand the definition of the difference quotient, let’s explore the formula.
Finding f (x + h)
The first step necessary to finding the difference quotient is to find our f (x + h). When working with a function, all you have to do is plug (x + h) into your function wherever you see an x.
Let’s look at the function: f (x) = 2x + 6.
To find our f (x + h), we need to plug in (x + h) into the function wherever we see an x.
f (x + h) = 2x + 6
Once we have our (x + h) plugged into the function, we must simplify our expression. In this case, we multiplied everything inside the parentheses by two to get:
F(x+h) = 2x + 2h + 6
Let’s look at a more difficult function: f (x) = 3x^2 + 4.
Again, we must plug (x + h) into the function wherever we see an x.
Plugging in (x + h) to the f (x + h)
Now, to simplify the expression, we must use the FOIL method to expand our (x + h)^2. Remember, FOIL stands for:
Multiply the first numbers: 3(x+h)^2
Multiply the outer numbers: 3(x+h) (x+h)
Multiply the inner numbers: 3 (x^2 +xh+xh+h^2)
Multiply the last numbers: 3 (x^2+2xh+h^2)
FOIL for (x + h)^2
We then plug our foiled expression back into the function to get:
Plugging the FOIL back in
To finish simplifying the expression, we multiply everything inside the parentheses by three.
Solution for f (x + h)
Not too difficult so far, right?
Finding the Difference Quotient
Now that we understand how to find f (x + h), we can plug our values into the difference quotient formula and simplify from there.
Let’s use our earlier example of f (x) = 3x^2 + 4.
We can plug in the expression we found for f (x + h) and our expression for f (x) to get a difference quotient of:
DQ with f (x + h) and f (x) plugged in
It is extremely important to keep each segment of the difference quotient inside of it’s own set of parentheses. This means that f (x + h) should be inside of it’s own set of parentheses and f (x) should be inside of it’s own set of parentheses.
Step Simplifying & Solving
This step is where some students make mistakes when working with the difference quotient. It is important to take note of the subtraction sign between f (x + h) and f (x). This subtraction sign tells us to change the sign in everything inside of the parentheses to the right of it. This allows us to get rid of our parentheses.
Removing the parentheses
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