Activity 8.1.4

We continue to work with $f(x)=e^x$ and the four approximations of degree $1$, $2$, $3$, and $4$ given by $T_1(x)$, $T_2(x)$, $T_3(x)$, and $T_4(x)$.

1. In Preview Activity, 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.

   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:

   Comparing $f(x) = e^x$ and its degree $1$, $2$, $3$, $4$ and approximations near $a = 0$.

   $\Delta x$	$x$	$f(x)$	$T_1(x)$	$T_2(x)$	$T_3(x)$	$T_4(x)$
   $0.1$	$-1.0$	$0.36787$	$0.00000$	$0.50000$	$0.33333$	$0.37500$
   $0.1$	$-0.9$	$0.40657$	$0.10000$	$0.50500$	$0.38350$	$0.41083$

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

   The absolute error between $f(x) = e^x$ and its degree $1$, $2$, $3$, and $4$ approximations at $x=-1$ and $x=-0.9$.

   $|f(x)-T_1(x)|$	$|f(x)-T_2(x)|$	$|f(x)-T_3(x)|$	$|f(x)-T_4(x)|$
   $0.36787$	$0.13212$	$0.03454$	$0.00712$
   $0.30657$	$0.09843$	$0.02307$	$0.00426$

2. 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 $x = 1$?

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

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

5. 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?

6. 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?