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.
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.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text: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.
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