§8.1 Extending local linearization

Introduction

Finding a quadratic approximation

Over and over again

Consider the function $f(x) = e^x$ near $a = 0$. We know that $f'(x) = e^x$, so $f'(0) = 1$; along with the fact that $f(0) = 1$, it follows that the tangent line approximation is

.

Build a spreadsheet that computes the difference between $f(x)$ and $L(x)$ for $x$-values between $-1$ and $1$, spaced $0.1$ units apart. Note: we will revisit this spreadsheet for additional computations in Activity 8.1.3, so it would be ideal if you save your work for later reference.

Your spreadsheet should start like the one shown in Table.

Note that since $b_0$, $b_1$, and $b_2$ are constants, if we take the derivative of the quadratic function $T_2$ using the sum and constant multiple rules, it follows that $T_2'(x) = b_1 + 2 b_2 x$.

What is $T_2''(x)$?

Recall that $f(x) = e^x$. Determine $f'(x)$ and $f''(x)$.

Enter the formulas you've determined for $f'(x)$, $f''(x)$, and $T_2''(x)$ to fill in the blanks below.

Next, observe that since $T_2(x) = b_0 + b_1 x + b_2 x^2$, it follows that $T_2(0) = b_0$. Reason similarly to determine the values of $T_2'(0)$ and $T_2''(0)$, as well as those of $f(0)$, $f'(0)$, and $f''(0)$ and enter these values appropriately in the blanks below.

Now, recall that we want the function values, first derivative values, and second derivative values of $f$ and $T_2$ to match at $a = 0$. What does $T_2(0) = f(0)$ tell us about the value of $b_0$, and what is its value? What does $T_2'(0) = f'(0)$ imply the value of $b_1$ is? How can we reason similarly to find $b_2$?

Having now determined the numerical values of $b_0$, $b_1$, and $b_2$, use appropriate computing technology to plot the function $T_2(x) = b_0 + b_1 x + b_2 x^2$ along with $f(x)=e^x$ and $T_1(x)=1+x$ in the same window as that shown in Figure.

What do you notice? For about which values of $x$ is $|f(x)-T_2(x)| \lt 0.1$?

By computing the third derivative of $f(x)$ and the second and third derivatives of $T_3(x)$ and evaluating the relevant functions at $x = 0$, fill in the blanks below.

Next, recall that we want $f$ and $T_3$ to share the same function and derivative values at $a = 0$ up to and including the third derivative. For instance, one of the four needed equations is $T_3'(0) = f'(0)$. Use the four equations your work in the preceding question to determine the values of $c_0$, $c_1$, $c_2$, and $c_3$.

Having now determined the numerical values of $c_0$, $c_1$, $c_2$, and $c_3$, use appropriate computational technology to plot the function $T_3(x) = c_0 + c_1 x + c_2 x^2 + c_3 x^3$ along with $f(x)=e^x$, $T_1(x)=1+x$, and $T_2(x) = 1 + x + \frac{1}{2}x^2$ in the same window as shown in Figure.

What do you notice? For approximately which values of $x$ is $|f(x)-T_3(x)| \lt 0.1$?

What if we wanted a degree-$4$ polynomial approximation to $f(x) = e^x$ near $a = 0$? Based on the patterns you've observed in $T_1$, $T_2$, and $T_3$, conjecture values for the constants $d_0, \ldots, d_4$ for a function $T_4$ of the form

that satisfies $T_4(0) = f(0)$, $T_4'(0) = f'(0)$, $\ldots$, $T_4^{(4)}(0) = f^{(4)}(0)$. Add this function $T_4$ to your plot in part (c) that includes $f(x)$ and the lower-degree polynomial approximations. What do you notice?