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

Verified step by step guidance

1. First, identify the function given: f(x) = x⁻¹. This means f(x) = 1/x.
2. 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).
3. Next, find the derivative of the function, f'(x). For f(x) = 1/x, the derivative f'(x) = -1/x².
4. Evaluate the derivative at x₀: f'(0.5) = -1/(0.5)².
5. 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|.

Key Concepts

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.

Approximation Error

The approximation error, |Δf - df|, quantifies the difference between the actual change in the function and the estimated change provided by the differential. This error helps assess the accuracy of the linear approximation and is important for understanding the limitations of using derivatives for estimating function values over intervals.