Extending local linearization

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 d…

Throughout our work in Section, 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 $a = 0$.

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

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

3. Use your work so far to determine the formula for $T_1(x)$, the tangent line approximation to $f(x)$…

In this exercise, we extend our work in Exercise. 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 $a = 0$.

1. Let $T_3(x) = k_0 + k_1x + k_2x^2 + k_3x^3$ and determine $T_3'(x)$, $T_3''(x)$, and $T_3'''(x)$. In addition, determine $f'''(x)$. Record your results below, along with the values of each of these functions at $a = 0$.

2. Use your work in above to find the formula for $T_3(x)$ that satisfies $T_3(0) = f(0)$, $T_3'(0) = f'(0)$, $T_3''(0) = f''(0)$, and $T_3'''(0) = f'''(0)$.

3. Plot $f(x)$, $T_1(x)$, $T_2(x)$, and $T_3(x)$ on the sam…