{Use of Tech} Maximum error Use the remainder term to find a bound on the error in the following approximations on the given interval. Error bounds are not unique.

√(1+x) ≈ 1 + x/2 on [−0.1,0.1]

검증된 단계별 안내

Identify the function to approximate: \(f(x) = \sqrt{1+x}\).

Recognize that the given approximation \(1 + \frac{x}{2}\) is the first-degree Taylor polynomial of \(f(x)\) centered at \(x=0\).

Recall the Lagrange form of the remainder (error) term for the Taylor polynomial of degree 1: \(R_1(x) = \frac{f^{\prime\prime}(c)}{2!} x^2\) for some \(c\) between 0 and \(x\).

Compute the second derivative of \(f(x)\): first, \(f'(x) = \frac{1}{2\sqrt{1+x}}\), then \(f''(x) = -\frac{1}{4(1+x)^{3/2}}\).

Find the maximum absolute value of \(f''(c)\) on the interval \([-0.1, 0.1]\) to bound the error, then use it in the remainder formula \(|R_1(x)| \leq \frac{\max|f''(c)|}{2} x^2\) to get the error bound.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Taylor Polynomial Approximation

A Taylor polynomial approximates a function near a point using derivatives at that point. For √(1+x), the linear approximation at x=0 is 1 + x/2, derived from the function's value and first derivative. This approximation is accurate close to the center point.

Remainder Term (Lagrange Error Bound)

The remainder term estimates the error between the actual function and its Taylor polynomial. The Lagrange form uses the (n+1)th derivative evaluated at some point in the interval to bound the error, ensuring the approximation's accuracy within a specified range.

Error Bound on a Closed Interval

To find the maximum error on [−0.1,0.1], evaluate the maximum absolute value of the relevant derivative on this interval. This maximum derivative value, combined with the remainder formula, provides a guaranteed upper bound on the approximation error.

{Use of Tech} Maximum error Use the remainder term to find a bound on the error in the following approximations on the given interval. Error bounds are not unique.√(1+x) ≈ 1 + x/2 on [−0.1,0.1]