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.
The function \(f(x)=\ln(x)\) graphed together with its degree \(1\) Taylor polynomial approximation near the point \((2,f(2))\text{.}\)
The graph shows that the function and its Taylor approximation intersect at the point \((2,f(2))\) and have the same slope at that point.
The two functions are very close together on the interval \((1.5,2.5)\text{,}\) but outside of that interval there start to be visual differences as the tangent line continues straight, but the function \(\ln(x)\) is curved.
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?
