Activity 8.2.3.
Let \(f(x) = \ln(x)\text{,}\) and recall that \(f\) is only defined for \(x \gt 0\text{.}\) As such, we can’t consider the tangent line (or any other) approximation at \(a = 0\text{.}\) Instead, we choose to work with an approximation to \(f(x) = \ln(x)\) centered at \(a = 1\) and will find the degree \(4\) Taylor polynomial approximation
\begin{equation*}
T_4(x) = c_0 + c_1 (x-1) + c_2 (x-1)^2 + c_3(x-1)^3 + c_4(x-1)^4\text{.}
\end{equation*}
(a)
Determine \(f'(x)\text{,}\) \(f''(x)\text{,}\) \(f'''(x)\text{,}\) and \(f^{(4)}(x)\text{,}\) and then compute \(f'(1)\text{,}\) \(f''(1)\text{,}\) \(f'''(1)\text{,}\) and \(f^{(4)}(1)\text{.}\) Enter your results in the provided blanks below.
\begin{align*}
k \amp= 0 \amp f(x) \amp= \ln(x) \amp f(1) \amp= \ln(1) = 0\\
k \amp= \fillinmath{XX} \amp f'(x) \amp= \fillinmath{XXXXX} \amp f'(1) \amp= \fillinmath{XXXXX}\\
k \amp= \fillinmath{XX} \amp f''(x) \amp= \fillinmath{XXXXX} \amp f''(1) \amp= \fillinmath{XXXXX}\\
k \amp= \fillinmath{XX} \amp f'''(x) \amp= \fillinmath{XXXXX} \amp f'''(1) \amp= \fillinmath{XXXXX}\\
k \amp= \fillinmath{XX} \amp f^{(4)}(x) \amp= \fillinmath{XXXXX} \amp f^{(4)}(1) \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
\begin{equation*}
T_4(x) = c_0 + c_1 (x-1) + c_2 (x-1)^2 + c_3(x-1)^3 + c_4(x-1)^4\text{.}
\end{equation*}
(c)
Use appropriate technology to plot \(f(x) = \ln(x)\text{,}\) its tangent line, \(T_1(x) = x - 1\text{,}\) and \(T_4(x)\) in the same window shown in Figure 8.2.17.
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 in the variable \(b\) to experiment with \(|f(b) - T_4(b)|\)); 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 your work in parts (a) and (b) to conjecture formulas for \(T_5(x)\) and \(T_6(x)\text{.}\)
For approximately 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 situation different from what we observed with \(f(x) = \cos(x)\) in Activity 8.2.2?
