Approximation Error

In Exercises 29–34, each function f(x) changes value when x changes from x₀ to x₀ + dx. Find

a. the change Δf = f(x₀ + dx) − f(x₀);
b. the value of the estimate df = fʹ(x₀) dx; and
c. the approximation error |Δf − df|.

f(x) = x⁻¹, x₀ = 0.5, dx = 0.1

검증된 단계별 안내

First, identify the function given: f(x) = x⁻¹. This means f(x) = 1/x.

Calculate the change in the function, Δf = f(x₀ + dx) − f(x₀). Substitute x₀ = 0.5 and dx = 0.1 into the function: Δf = f(0.5 + 0.1) − f(0.5).

Next, find the derivative of the function, f'(x). For f(x) = 1/x, the derivative f'(x) = -1/x².

Evaluate the derivative at x₀: f'(0.5) = -1/(0.5)².

Calculate the estimate df = f'(x₀) dx using the derivative value found: df = f'(0.5) * 0.1. Finally, find the approximation error |Δf − df|.

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.

Change in Function (Δf)

The change in a function, denoted as Δf, represents the actual difference in the function's value as the input changes from x₀ to x₀ + dx. It is calculated as Δf = f(x₀ + dx) - f(x₀). This concept is crucial for understanding how the function behaves over a small interval and serves as the basis for approximating changes using derivatives.

Differential (df)

The differential, df, is an estimate of the change in the function based on the derivative at a specific point, multiplied by a small change in the input, dx. It is expressed as df = f'(x₀) dx. This concept is essential for approximating the change in the function using linearization, which simplifies calculations in calculus.