Activity 8.2.2.
Let \(f(x) = \cos(x)\text{.}\) Through the questions that follow, we seek to find the degree \(n\) Taylor polynomial for \(f(x)\) centered at \(a = 0\text{.}\)
(a)
Determine the first \(8\) derivatives of \(f(x) = \cos(x)\) and evaluate each at \(a = 0\text{.}\) Summarize your work by filling in all the blanks below.
\begin{align*}
k \amp= 0 \amp f(x) \amp= \cos(x) \amp f(0) \amp= \cos(0) = 1 \\
k \amp= \fillinmath{XX} \amp f'(x) \amp= \fillinmath{XXXXX} \amp f'(0) \amp= \fillinmath{XXXXX} \\
k \amp= \fillinmath{XX} \amp f''(x) \amp= \fillinmath{XXXXX} \amp f''(0) \amp= \fillinmath{XXXXX} \\
k \amp= \fillinmath{XX} \amp f'''(x) \amp= \fillinmath{XXXXX} \amp f'''(0) \amp= \fillinmath{XXXXX} \\
k \amp= \fillinmath{XX} \amp f^{(4)}(x) \amp= \fillinmath{XXXXX} \amp f^{(4)}(0) \amp= \fillinmath{XXXXX} \\
k \amp= \fillinmath{XX} \amp f^{(5)}(x) \amp= \fillinmath{XXXXX} \amp f^{(5)}(0) \amp= \fillinmath{XXXXX} \\
k \amp= \fillinmath{XX} \amp f^{(6)}(x) \amp= \fillinmath{XXXXX} \amp f^{(6)}(0) \amp= \fillinmath{XXXXX} \\
k \amp= \fillinmath{XX} \amp f^{(7)}(x) \amp= \fillinmath{XXXXX} \amp f^{(7)}(0) \amp= \fillinmath{XXXXX} \\
k \amp= \fillinmath{XX} \amp f^{(8)}(x) \amp= \fillinmath{XXXXX} \amp f^{(8)}(0) \amp= \fillinmath{XXXXX}
\end{align*}
(b)
Use your work in (a) along with the fact that the coefficients of the Taylor polynomial are determined by \(c_k = \frac{f^{(k)}(0)}{k!}\) to find formulas for \(T_2(x)\text{,}\) \(T_4(x)\text{,}\) \(T_6(x)\text{,}\) and \(T_8(x)\text{.}\)
(c)
Based on the patterns you observe in your prior work, what do you expect to be the formula for \(T_{10}(x)\text{?}\)
(d)
Use appropriate computing technology to plot \(T_4(x)\) and \(T_6(x)\) along with \(f(x) = \cos(x)\) and \(T_2(x)\) in the same window as shown in Figure 8.2.9.
What do you notice?
(e)
Build a spreadsheet similar to the one in Table 8.1.9 and Table 8.1.10 from Activity 8.1.4, but do so using \(\Delta x = 0.2\text{,}\) a start value of \(x = -2\text{,}\) and the functions \(f(x) = \cos(x)\text{,}\) \(T_2(x)\text{,}\) \(T_4(x)\text{,}\) and \(T_6(x)\text{.}\) The first six columns of your spreadsheet should begin as shown in Table 8.2.12.
| \(\Delta x\) | \(x\) | \(f(x)\) | \(T_2(x)\) | \(T_4(x)\) | \(T_6(x)\) |
| \(0.2\) | \(-2.0\) | \(-0.41615\) | \(-1.00000\) | \(-0.33333\) | \(-0.42222\) |
| \(0.2\) | \(-1.8\) | \(-0.22720\) | \(-0.62000\) | \(-0.18260\) | \(-0.22984\) |
and the last three columns of your spreadsheet should begin as follows:
| \(|f(x)-T_2(x)|\) | \(|f(x)-T_4(x)|\) | \(|f(x)-T_6(x)|\) |
| \(0.58385\) | \(0.08281\) | \(0.00608\) |
| \(0.39280\) | \(0.04460\) | \(0.00263\) |
For about what interval of \(x\)-values is it true that \(|f(x) - T_2(x)| \lt 0.1\text{?}\) How does the interval of \(x\)-values change if we instead consider where \(|f(x) - T_4(x)| \lt 0.1\text{?}\) \(|f(x) - T_6(x)| \lt 0.1\text{?}\)
