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 In(2). 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). a. Find the tangent line to f at x = = 0 and use this linearization to approximate In(2). That is, find L(x), the tangent line approximation to f(x), and use the fact that L(1) ≈ ƒ(1) to estimate In(2). b. The linearization of In(1 + x) does not provide a very good approximation to ln(2) since 1 is not that close to 0. To obtain a better approximation, we alter our approach; instead of using a straight line to approximate In(2), we use a quadratic function to account for the concavity of In(1 + x) for x close to 0. With the linearization, both the function's value and slope agree with the linearization's value and slope at x 0. We will now make a quadratic approximation P₂(x) to ƒ(x) = ln(1 + x) centered at a with the property that P₂(0) = f(0), P₂(0) = f'(0), and P"(0) = f'(0). = = 0 i. Let P₂(x): = x - 2. Show that P₂(0) = ƒ(0), P₂(0) = ƒ'(0), and P" (0) = f'(0). Use P₂(x) to approximate In (2) by using the fact that P₂(1) ≈ ƒ(1). ii. We can continue approximating In(2) with polynomials of larger degree whose derivatives agree with those of f at 0. This makes the polynomials fit the graph of f better for more values of around 0. For example, let P3(x) = x + Show that P3(0) = f(0), P²(0) = f'(0), Pg(0) = f'(0), and - P" (0) = f'(0). Taking a similar approach to preceding questions, use P³(x) to approximate ln(2). iii. If we used a degree 4 or degree 5 polynomial to approximate In(1 + x), what approximations of In(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.

25. Figure 12 shows two populations P1 and P2. Assuming they are exponentially growth populations, (a) which population has the largest doubling time. Show your rationale. (b) Calculate at what time P2 will be 25.6 times P1? Population 4004 300 200 100 P1 Figure 12 10 P₂ 15 time (years)

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Help me solve question #20