§8.1 Extending local linearization (continued)

As the degree of the approximation increases

Summary

In Preview Activity 8.1, we built a spreadsheet that computed the differences between $f(x)$ and $T_1(x)$ for $x$-values between $-1$ and $1$, spaced $0.1$ units apart. Your spreadsheet started like the one shown in the table in Preview Activity 8.1.

Next, we build an updated version of this spreadsheet that computes similar differences between $f$ and the three higher degree approximations we have found. In particular, we now want to have columns for $\Delta x$, $x$, $f(x)$, $T_1(x)$, $T_2(x)$, $T_3(x)$, and $T_4(x)$, plus the absolute differences $|f(x) - T_1(x)|$, $|f(x) - T_2(x)|$, $|f(x) - T_3(x)|$, and $|f(x) - T_4(x)|$. Hint: when building your entries, note that you can think of $T_2(x)$ as $T_2(x) = T_1(x) + \frac{1}{2}x^2$, and similarly view $T_3(x)$ as “$T_2(x)$ plus one more term”.

Include at least $5$ digits of accuracy beyond the decimal. The first seven columns of your spreadsheet might start like this:

$\Delta x$ | $x$ | $f(x)$ | $T_1(x)$ | $T_2(x)$ | $T_3(x)$ | $T_4(x)$

$0.1$ | $-1.0$ | $0.36787$ | $0.00000$ | $0.50000$ | $0.33333$ | $0.37500$

$0.1$ | $-0.9$ | $0.40657$ | $0.10000$ | $0.50500$ | $0.38350$ | $0.41083$

The next four columns of your spreadsheet should begin as follows:

$|f(x)-T_1(x)|$ | $|f(x)-T_2(x)|$ | $|f(x)-T_3(x)|$ | $|f(x)-T_4(x)|$

$0.36787$ | $0.13212$ | $0.03454$ | $0.00712$

$0.30657$ | $0.09843$ | $0.02307$ | $0.00426$

We call the value of $|f(x) - T_2(x)|$ the absolute error of the quadratic approximation of $f$ at the value $x$. What is the absolute error of the quadratic approximation at $x = -1$? at $x = 1$?

What is the absolute error of the cubic (degree $3$) approximation, $T_3(x)$, at $x = -1$? at $x = 1$?

What is the absolute error of the quartic (degree $4$) approximation, $T_4(x)$, at $x = -1$? at $x = 1$?

Study your spreadsheet for trends that you notice as the value of $x$ changes or the degree $n$ of the approximation changes. What are your observations?

Investigate the errors in the various approximations for a wider interval of $x$-values. For example, you might consider starting at $x = -2$ with $\Delta x = 0.2$. What do you notice?

Throughout our work in Section 8.1, we have focused on approximating the function $f(x) = e^x$. In this exercise, we change the function of interest to $f(x) = \frac{1}{3}x^3 + \frac{1}{4}x^2 - 2x - 1$, and consider the linear and quadratic approximations to $f$ near $a = 0$.

• Determine $f'(x)$ and $f''(x)$ and enter their formulas below.

• Next, compute $f(0)$, $f'(0)$, and $f''(0)$ and enter those values below.

• Use your work so far to determine the formula for $T_1(x)$, the tangent line approximation to $f(x)$ at $a =0$ (which satisfies $T_1(0) = f(0)$ and $T_1'(0) = f'(0)$). - Let $T_2(x)$ be the quadratic approximation to $f(x)$ near $a = 0$ that satisfies $T_2(0) = f(0)$, $T_2'(0) = f'(0)$, and $T_2''(0) = f''(0)$. You might start by letting $T_2(x) = c_0 + c_1 x + c_2 x^2$, and creating an updated table like the one shown below.

Use your work in in the table above to find the formula for $T_2(x)$ that satisfies $T_2(0) = f(0)$, $T_2'(0) = f'(0)$, and $T_2''(0) = f''(0)$. - Plot $f(x)$, $T_1(x)$, and $T_2(x)$ on the same axes, centered at $a = 0$. What do you notice?

In this exercise, we extend our work in Exercise 8.1.1. We continue to consider the function $f(x) = \frac{1}{3}x^3 + \frac{1}{4}x^2 - 2x - 1$, but now build the cubic (degree $3$) approximation to $f$ near $a = 0$.