Chapter 8: Q. 5 (page 679)

Let f be a twice-differentiable function at a point $x_{0}$ . Using the words value, slope, and concavity, explain why the second Taylor polynomial $P_{2} (x)$ might be a good approximation for f close to $x_{0}$ .

Short Answer

Expert verified

$The second Taylor polynomial has the same value, same slope for its tangent line, and the same concavity as the function f at the point x_{0} .$

Step by step solution

Step 1. Given information is:

$f is a twice - differentiable function at a point x_{0}$

Step 2. Result

$If f is a twice differentiable function at a point x_{0}, then the second Taylor polynomial P_{2} (x) might be a good approximation for f close to x_{0} . This is because the second Taylor polynomial has the same value, same slope for its tangent line, and the same concavity as the function f at the point x_{0} .$

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Let f be a twice-differentiable function at a point $x_{0}$ . Using the words value, slope, and concavity, explain why the second Taylor polynomial $P_{2} (x)$ might be a good approximation for f close to $x_{0}$ .

Short Answer

Step by step solution

Step 1. Given information is:

Step 2. Result

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Let f be a twice-differentiable function at a point x0. Using the words value, slope, and concavity, explain why the second Taylor polynomial P2(x)might be a good approximation for f close to x0.

Short Answer

Step by step solution

Step 1. Given information is:

Step 2. Result

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

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Study anywhere. Anytime. Across all devices.

Let f be a twice-differentiable function at a point $x_{0}$ . Using the words value, slope, and concavity, explain why the second Taylor polynomial $P_{2} (x)$ might be a good approximation for f close to $x_{0}$ .