Exercise 8

In Exercise, we found that the degree 2 Taylor polynomial centered at $a = 0$ 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.

Let $f(x) = \frac{1}{2}x^2 - 2x + 5$, and let $a = 2$ be the center at which we will find a degree $2$ Taylor polynomial approximation of $f$.

1. By finding $f'(x)$, $f''(x)$, $f(2)$, $f'(2)$, and $f''(2)$, determine $T_2(x)$, the degree $2$ Taylor polynomial approximation of $f$ that is centered at $a = 2$.

2. Plot both $f(x)$ and $T_2(x)$ on the same axes. What do you observe?

3. What does the algebraic form of $T_2(x)$ tell you about the original function $f(x)$?

4. What is the tangent line approximation to $f(x)$ at $a = 2$? What is special about the function's behavior at this input value?