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Chapter 9: Problem 1

Suppose you use a second-order Taylor polynomial centered at 0 to approximate a function \(f\). What matching conditions are satisfied by the polynomial?

Short Answer

Expert verified
Answer: The matching conditions satisfied by a second-order Taylor polynomial centered at 0 for approximating a function f are: 1. The polynomial must have the same function value at 0: \(f(0) = P_2(0)\). 2. The polynomial must have the same first derivative at 0: \(f'(0) = P_2'(0)\). 3. The polynomial must have the same second derivative at 0: \(f''(0) = P_2''(0)\).

Step by step solution

01

Matching Condition 1: Function Value at 0

The second-order Taylor polynomial must match the function at the center, 0. In other words, \(f(0) = P_2(0)\).
02

Matching Condition 2: First Derivative Value at 0

The first derivative of \(P_2(x)\) is given by: \(P_2'(x) = f'(0) + f''(0)x\) At the center, 0, the first derivative of the second-order Taylor polynomial must match the first derivative of the function, which means \(f'(0) = P_2'(0)\).
03

Matching Condition 3: Second Derivative Value at 0

Taking the second derivative of the polynomial \(P_2(x)\), we get: \(P_2''(x) = f''(0)\) In this case, the second derivative is constant and does not depend on x. The second derivative of \(P_2(x)\) must match the second derivative of the function at the center, 0. Thus, \(f''(0) = P_2''(0)\). To recap, the matching conditions satisfied by a second-order Taylor polynomial centered at 0 for approximating a function \(f\) are: 1. The polynomial must have the same function value at 0: \(f(0) = P_2(0)\). 2. The polynomial must have the same first derivative at 0: \(f'(0) = P_2'(0)\). 3. The polynomial must have the same second derivative at 0: \(f''(0) = P_2''(0)\).

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