Activity 8.2.4.
This activity builds on Activity 8.2.3, and only changes one key thing: the location where the approximation is centered. Again, we let \(f(x) = \ln(x)\text{,}\) and recall that \(f\) is only defined for \(x \gt 0\text{.}\) Here, we choose to work with an approximation centered at \(a=2\text{,}\) and find the degree \(4\) Taylor polynomial approximation
\begin{equation*}
T_4(x) = c_0 + c_1 (x-2) + c_2 (x-2)^2 + c_3(x-2)^3 + c_4(x-2)^4\text{.}
\end{equation*}
(a)
We recall \(f'(x)\text{,}\) \(f''(x)\text{,}\) \(f'''(x)\text{,}\) and \(f^{(4)}(x)\) from our work in Activity 8.2.3, and then compute \(f'(2)\text{,}\) \(f''(2)\text{,}\) \(f'''(2)\text{,}\) and \(f^{(4)}(2)\text{.}\) Enter the updated results in the blanks below.
\begin{align*}
k \amp= 0 \amp f(x) \amp= \ln(x) \amp f(2) \amp= \fillinmath{XXXXX}\\
k \amp= 1 \amp f'(x) \amp= x^{-1} \amp f'(2) \amp= \fillinmath{XXXXX}\\
k \amp= 2 \amp f''(x) \amp= (-1)x^{-2} \amp f''(2) \amp= \fillinmath{XXXXX}\\
k \amp= 3 \amp f'''(x) \amp= (-2)(-1)x^{-3} \amp f'''(2) \amp= \fillinmath{XXXXX}\\
k \amp= 4 \amp f^{(4)}(x) \amp= (-3)(-2)(-1)x^{-4} \amp f^{(4)}(2) \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 \(c_k = \frac{f^{(k)}(a)}{k!}\) to determine \(T_4(x) = c_0 + c_1 (x-2) + c_2 (x-2)^2 + c_3(x-2)^3 + c_4(x-2)^4\text{.}\)
(c)
Use appropriate technology to plot \(f(x) = \ln(x)\text{,}\) its tangent line, \(T_1(x) = \ln(2) + \frac{1}{2}(x - 2)\text{,}\) and \(T_4(x)\) on the same axes in the window shown Figure 8.2.18.
What do you notice?
(d)
Compute \(|f(x) - T_4(x)|\) for several different \(x\) values (you might find it helpful to use a slider in Desmos); for approximately what values of \(x\) is it true that \(|f(x) - T_4(x)| \lt 0.1\text{?}\)
(e)
Use the patterns you observe in parts (a) and (b) to conjecture formulas for \(T_5(x)\) and \(T_6(x)\text{.}\)
For about what interval of \(x\)-values is it true that \(|f(x) - T_5(x)| \lt 0.1\text{?}\) What about \(|f(x) - T_6(x)| \lt 0.1\text{?}\) How is this different from what we observed with the Taylor approximations centered at \(a = 1\) in Activity 8.2.3? How is it similar?
