In this section, we will learn how we can more systematically find degree approximations for functions such as that have at least derivatives, as well as how to center the approximation at a value other than .
Moreover, we found that as the degree of the approximation increased, the polynomial approximation got more accurate by being closer to at each fixed value of as well as on a wider interval.
We can also notice that the linear approximation is embedded within the quadratic approximation, and the quadratic approximation within the cubic one, and so on. These observations naturally lead us to consider approximations of arbitrary degree in order to generate more accurate approximations for any function with a sufficient number of derivatives. We thus define the Taylor polynomial of degree centered at .
enable us to determine the coefficients in terms of the values of the various derivatives of . First, we take derivatives of , and assemble those in Table 8.2.4. As we do so, we choose not to combine products of numbers that arise in order to see certain patterns in the coefficients.
Next, we (1) evaluate each of the derivatives at , and then (2) set the result equal to the corresponding derivative value of evaluated at , which ultimately enables us to determine the coefficients . These two steps are summarized in Table 8.2.5.
We see a natural pattern that results from taking the th derivative of a degree term. For example, the repeated derivatives of are ,,, and finally . By the time we get to the fourth derivative of , only a constant remains, and that constant is the factorial 2
For any positive whole number , its factorial, is the product of all of the positive whole numbers less than or equal to :.
From the rightmost column of Table 8.2.5, we now see how the values of are determined by the values of the various derivatives evaluated at , each scaled by a corresponding factorial. In particular, solving each equation in the rightmost column of Table 8.2.5 for , we see that
This enables us to find the Taylor polynomial of degree for any function by finding the values of and using these numbers to determine . We summarize our recent work as follows.
To find the degree Taylor polynomial, we need to compute ,,,,, so we first find the first through fifth derivatives of in the left column of Table 8.2.7, and then evaluate those derivatives at in the right column.
Table8.2.7.Finding the derivatives of at .
When finding the coefficients of a Taylor polynomial, it is often helpful to not combine products such as and into a single number, in order to better observe patterns; indeed, by not combining the constants that arise in higher derivatives of , we see patterns of alternating signs and factorials that arise. From the right column of Table 8.2.7 and the fact that , we see that
so that the degree Taylor approximation of at is
.
Thus, we have found the approximation
,
and plotting along with and in Figure 8.2.8, we see how much better the degree approximation is than the tangent line approximation.
Figure8.2.8.The function and its degree Taylor approximation near the point , along with .
Determine the first derivatives of and evaluate each at . Summarize your work by filling in all the blanks in Table 8.2.9.
Table8.2.9.Finding the derivatives of at .
Use your work in (a) along with the fact that the coefficients of the Taylor polynomial are determined by to find .
Based on the pattern you observe in Table 8.2.9, what do you expect to be the formula for ?
In Figure 8.2.10, we see and plotted on the same axes. Add and to the figure by plotting those two functions on along with and . What do you notice?
Figure8.2.10.The function and its degree Taylor approximation near the point .
Build a spreadsheet similar to the one in Table 8.1.12 and Table 8.1.13 from Activity 8.1.4, but do so using , a start value of , and the functions ,,, and . The first six columns of your spreadsheet should begin as shown in Table 8.2.11,
Table8.2.11.Comparing and its degree ,, and approximations near .
and the last three columns of your spreadsheet should begin as follows:
Table8.2.12.The absolute error between and its degree ,, and approximations.
For about what interval of -values is it true that ? How does the interval of -values change if we instead consider where ??
using the degree Taylor approximation of the sine function at . Furthermore, as we saw in Section 8.1, the degree Taylor approximation of provides us with the estimate
Thus, we now have higher degree Taylor approximations for ,, and that exhibit interesting patterns in their coefficients that we can use to easily find higher degree approximations that are even more accurate. Indeed, these approximations are what computational devices use to find numerical estimates for quantities such as ,, and . For example,
Subsection8.2.2Taylor polynomial approximations centered at an arbitrary value
In all of our work so far in Chapter 8, we have focused on approximating functions such as ,,, and near . But we could instead be interested in the behavior of some function near , or be interested in a function that wasn’t even defined at . Thus, we next consider how we can generalize our earlier work to Taylor polynomial approximations centered at any value .
that satisfies for values near . Provided that has a second derivative at , we can build a quadratic approximation near for , similar to the one we found at for in Activity 8.1.3. In addition, as long as has a third derivative at , we can even find a cubic approximation (just as we did at in Activity 8.1.4), and so on.
In developing such approximations centered at any value , our guiding principle is the same as with our work at : we’ll require that at the input value , the original function’s output and its derivatives’ outputs match the corresponding approximation’s output and derivatives’ output.
Similar to the situation when , it follows that we can find the coefficients of the Taylor polynomial in terms of the various derivatives of evaluated at .
Let , and recall that is only defined for . As such, we can’t consider the tangent line (or any other) approximation at . Instead, we choose to work with an approximation to centered at and will find the degree Taylor polynomial approximation
Determine ,,, and , and then compute ,,, and . Enter your results in in Table 8.2.22.
Table8.2.22.Finding the derivatives of at .
Use your work in (a) along with the fact that the coefficients of the Taylor polynomial are determined by to determine
.
In Figure 8.2.23, we see and its tangent line, plotted on the same axes. Add to the figure. What do you notice?
Figure8.2.23.The function and its degree Taylor approximation near the point .
Compute for several different values (you might find it helpful to use a slider in Desmos in the variable to experiment with ); for approximately what values of is it true that ?
Use the pattern you observe in Table 8.2.22 to conjecture formulas for and .
For about what interval of -values is it true that ? What about ? How is this different from what we observed with in Activity 8.2.2?
This activity builds on Activity 8.2.3, and only changes one key thing: the location where the approximation is centered. Again, we let , and recall that is only defined for . Here, we choose to work with an approximation centered at , and find the degree Taylor polynomial approximation
Use your work in (a) along with the fact that the coefficients of the Taylor polynomial are determined by to determine .
In Figure 8.2.29, we see and its tangent line, plotted on the same axes. Add to the figure. What do you notice?
Figure8.2.29.The function and its degree Taylor approximation near the point .
Compute for several different values (you might find it helpful to use a slider in Desmos); for approximately what values of is it true that ?
Use the pattern you observe in Table 8.2.28 to conjecture formulas for and .
For about what interval of -values is it true that ? What about ? How is this different from what we observed with the Taylor approximations centered at in Activity 8.2.3? How is it similar?
Provided that a function has derivatives at a selected input value , we can find a degree polynomial that approximates near by requiring that ,,,,.
When , the degree polynomial approximation, , to a function , centered at , is a polynomial of the form
and it follows that the coefficients are determined by the values of the various derivatives of evaluated at according to the formula
Moreover, for near ,
.
Just as we can consider any function that has derivatives at and find approximations centered there, we can also consider any input value at which those derivatives exist, and find a polynomial approximation that satisfies ,,,,.
At such a value , the degree Taylor polynomial of centered at has form
and it follows that the coefficients are determined by the values of the various derivatives of evaluated at according to the formula
In Exercise 8.2.4.5, we found that the degree 2 Taylor polynomial centered at of a quadratic function is the quadratic function itself. In this exercise, we explore how changing the center of the approximation offers additional insight into the function.
Recall that we found in Preview 8.2.1 and subsequent work that , which is the degree Taylor approximation centered at . And in Activity 8.2.2, we found that the degree Taylor approximation centered at for is .
where is the degree Taylor approximation of centered at . (Here we are using “” and “” to distinguish between these two degree polynomial approximations of the two different functions and , centered at two different values.)
