A student will be able to:
- Extend the Mean Value Theorem to make linear approximations.
- Analyze errors in linear approximations.
- Extend the Mean Value Theorem to make quadratic approximations.
- Analyze errors in quadratic approximations.
In this lesson we will use the Mean Value Theorem to make approximations of functions. We will apply the Theorem directly to make linear approximations and then extend the Theorem to make quadratic approximations of functions.
Let’s consider the tangent line to the graph of a function at the point The equation of this line is We observe from the graph that as we consider near the value of is very close to
In other words, for values close to the tangent line to the graph of a function at the point provides an approximation of or We call this the linear or tangent line approximation of at and indicate it by the formula
The linear approximation can be used to approximate functional values that deviate slightly from known values. The following example illustrates this process.
Use the linear approximation of the function at to approximate .
We know that . So we will find the linear approximation of the function and substitute values close to
We note that
We also know that
By substitution, we have
We observe that to approximate we need to evaluate the linear approximation at , and we have
. If we were to compare this approximation to the actual value, , we see that it is a very good approximation.
If we observe a table of values close to we see how the approximations compare to the actual value.
Setting Error Estimates
We would like to have confidence in the approximations we make. We therefore can choose the values close to a to ensure that the errors are within acceptable boundaries. For the previous example, we saw that the values of close to gave very good approximations, all within of the actual value.
Let’s suppose that for the previous example, we did not require such precision. Rather, suppose we wanted to find the range of values close to that we could choose to ensure that our approximations lie within of the actual value.
The easiest way for us to find the proper range of values is to use the graphing calculator. We first note that our precision requirement can be stated as
If we enter the functions and into the menu as and , respectively, we will be able to view the function values of the functions using the [TABLE] feature of the calculator. In order to view the differences between the actual and approximate values, we can enter into the menu the difference function as follows:
- Go to the menu and place cursor on the line.
- Press the following sequence of key strokes: [VARS] [FUNCTION] . This will copy the function onto the line of the menu.
- Press [-] to enter the subtraction operation onto the line of the menu.
- Repeat steps 1 - 2 and choose to copy onto the line of the menu.
Your screen should now appear as follows:
Now let’s setup the [TABLE] function so that we find the required accuracy.
- Press 2ND followed by [TBLSET] to access the Table Setup screen.
- Set the [TBLStart] value to and to .
Your screen should now appear as follows:
Now we are ready to find the required accuracy.
Access the [TABLE] function, scroll through the table, and find those values that ensure .
It turns out that the linear approximations we have discussed are not the only approximations that we can derive using derivatives. We can use non-linear functions to make approximations. These are called Taylor Polynomials and are defined as
We call this the Taylor Polynomial of fcentered at a.
For our discussion, we will focus on the quadratic case. The Taylor Polynomial corresponding to is given by
Note that this is just our linear approximation with an added term. Hence we can view it as an approximation of for values close to
Find the quadratic approximation of the function at and compare them to the linear approximations from the first example.
So . If we update our table from the first example we can see how the quadratic approximation compares with the linear approximation.
As you can see from the graph below, is an excellent approximation of near
We get a slightly better approximation for the quadratic than for the linear. If we reflect on this a bit, the finding makes sense since the shape and properties of quadratic functions more closely approximate the shape of radical functions.
Finally, as in the first example, we wish to determine the range of values that will ensure that our approximations are within of the actual value. Using the [TABLE] feature of the calculator, we find that if then .
- We extended the Mean Value Theorem to make linear approximations.
- We analyzed errors in linear approximations.
- We extended the Mean Value Theorem to make quadratic approximations.
- We analyzed errors in quadratic approximations.
In problems #1–4, find the linearization of the function at
- Find the linearization of the function near a = 1 and use it to approximate .
- Based on using linear approximations, is the following approximation reasonable?
- Use a linear approximation to approximate the following:
- Verify the the following linear approximation at Determine the values of for which the linear approximation is accurate to
- Find the quadratic approximation for the function in #3, near
- Determine the values of for which the quadratic approximation found in #7 is accurate to
- Determine the quadratic approximation for near Do you expect that the quadratic approximation is better or worse than the linear approximation? Explain your answer.
- Yes; using linear approximation on near we find that ;
- Using linear approximation on near we find ;
; we would expect it to be a better approximation since the graph of is similar to the graph of a quadratic function.
Texas Instruments Resources
In the CK-12 Texas Instruments Calculus FlexBook, there are graphing calculator activities designed to supplement the objectives for some of the lessons in this chapter. See http://www.ck12.org/flexr/chapter/9728.