Q. 6 Let f be a twice-differentiable ... [FREE SOLUTION]

Chapter 8: Q. 6 (page 679)

Let f be a twice-differentiable function at a point $x_{0}$ . Explain why the sum
$f (x) + f' (x) (x - x_{0}) + \frac{f'' (x)}{2!} {(x - x_{0})}^{2}$
is not the second-order Taylor polynomial for f at $x_{0}$ .

Short Answer

Expert verified

$The given function is not a second - order Taylor polynomial for f at x_{0} . This is because the function f (x) and its derivatives, that is, f' (x) and f'' (x) are not evaluated at x = x_{0} .$

Step by step solution

Step 1. Given information is:

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

Step 2. Solving

$Consider g (x) = f (x) + f' (x) (x - x_{0}) + \frac{f'' (x)}{2!} {(x - x_{0})}^{2} The second order Taylor polynomial for f at x_{0} : f (x_{0}) + f' (x_{0}) (x - x_{0}) + \frac{f'' (x_{0})}{2!} {(x - x_{0})}^{2} The second order Taylor polynomial in terms of g is : g (x_{0}) = f (x_{0}) + f' (x_{0}) (x - x_{0}) + \frac{f'' (x_{0})}{2!} {(x - x_{0})}^{2} The function g (x) is not a second - order Taylor polynomial for f at x_{0} . This is because the function f (x) and its derivatives, that is, f' (x) and f'' (x) are not evaluated at x = x_{0} .$

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91影视

Let f be a twice-differentiable function at a point $x_{0}$ . Explain why the sum
$f (x) + f' (x) (x - x_{0}) + \frac{f'' (x)}{2!} {(x - x_{0})}^{2}$
is not the second-order Taylor polynomial for f at $x_{0}$ .

Short Answer

Step by step solution

Step 1. Given information is:

Step 2. Solving

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91影视

Let f be a twice-differentiable function at a point x0. Explain why the sumf(x)+f'(x)(x-x0)+f''(x)2!(x-x0)2is not the second-order Taylor polynomial for f at x0.

Short Answer

Step by step solution

Step 1. Given information is:

Step 2. Solving

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Probability and Statistics

Mechanics Maths

Applied Mathematics

Calculus

Decision Maths

Study anywhere. Anytime. Across all devices.

Let f be a twice-differentiable function at a point $x_{0}$ . Explain why the sum
$f (x) + f' (x) (x - x_{0}) + \frac{f'' (x)}{2!} {(x - x_{0})}^{2}$
is not the second-order Taylor polynomial for f at $x_{0}$ .