Preview Activity 8.4.1.
Preview Activity 8.3.1 showed how we can approximate the number \(e\) with linear, quadratic, and other polynomial approximations. We use a similar approach in this activity to obtain linear and quadratic approximations to \(\ln(2)\text{.}\) Along the way, we encounter a type of series that is different than most of the ones we have seen so far. Throughout this activity, let \(f(x) = \ln(1+x)\text{.}\)
(a)
Find the tangent line to \(f\) at \(x=0\) and use this linearization to approximate \(\ln(2)\text{.}\) That is, find \(L(x)\text{,}\) the tangent line approximation to \(f(x)\text{,}\) and use the fact that \(L(1) \approx f(1)\) to estimate \(\ln(2)\text{.}\)
(b)
The linearization of \(\ln(1+x)\) does not provide a very good approximation to \(\ln(2)\) since \(1\) is not that close to \(0\text{.}\) To obtain a better approximation, we alter our approach; instead of using a straight line to approximate \(\ln(2)\text{,}\) we use a quadratic function to account for the concavity of \(\ln(1+x)\) for \(x\) close to \(0\text{.}\) With the linearization, both the function’s value and slope agree with the linearization’s value and slope at \(x=0\text{.}\) We will now make a quadratic approximation \(P_2(x)\) to \(f(x) = \ln(1+x)\) centered at \(x=0\) with the property that \(P_2(0) = f(0)\text{,}\) \(P'_2(0) = f'(0)\text{,}\) and \(P''_2(0) = f''(0)\text{.}\) Let \(P_2(x) = x - \frac{x^2}{2}\text{.}\) Show that \(P_2(0) = f(0)\text{,}\) \(P'_2(0) = f'(0)\text{,}\) and \(P''_2(0) = f''(0)\text{.}\) Use \(P_2(x)\) to approximate \(\ln(2)\) by using the fact that \(P_2(1) \approx f(1)\text{.}\)
(c)
We can continue approximating \(\ln(2)\) with polynomials of larger degree whose derivatives agree with those of \(f\) at \(0\text{.}\) This makes the polynomials fit the graph of \(f\) better for more values of \(x\) around \(0\text{.}\) For example, let \(P_3(x) = x - \frac{x^2}{2}+\frac{x^3}{3}\text{.}\) Show that \(P_3(0) = f(0)\text{,}\) \(P'_3(0) = f'(0)\text{,}\) \(P''_3(0) = f''(0)\text{,}\) and \(P'''_3(0) = f'''(0)\text{.}\) Taking a similar approach to preceding questions, use \(P_3(x)\) to approximate \(\ln(2)\text{.}\)
(d)
If we used a degree \(4\) or degree \(5\) polynomial to approximate \(\ln(1+x)\text{,}\) what approximations of \(\ln(2)\) do you think would result? Use the preceding questions to conjecture a pattern that holds, and state the degree \(4\) and degree \(5\) approximation.
