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) = x⁴, x₀ = 1, dx = 0.1

Verified step by step guidance

1. Step 1: Calculate the change in the function, Δf = f(x₀ + dx) − f(x₀). For the function f(x) = x⁴, substitute x₀ = 1 and dx = 0.1 to find Δf = (1 + 0.1)⁴ - 1⁴.
2. Step 2: Find the derivative of the function, f'(x). For f(x) = x⁴, the derivative is f'(x) = 4x³. Evaluate this at x₀ = 1 to get f'(1) = 4(1)³.
3. Step 3: Calculate the estimate df = f'(x₀) dx. Using the derivative value from Step 2, df = 4(1)³ * 0.1.
4. Step 4: Determine the approximation error, |Δf − df|. Use the values of Δf from Step 1 and df from Step 3 to find the absolute difference.
5. Step 5: Interpret the results. The approximation error |Δf − df| gives an indication of how well the linear approximation (tangent line) estimates the actual change in the function over the interval from x₀ to x₀ + dx.

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 comparing actual changes to approximations.

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 x (dx). It is expressed as df = f'(x₀) dx. This concept is essential for approximating how a function changes locally and is foundational in calculus for understanding rates of change.

Approximation Error

Approximation error quantifies the difference between the actual change in the function (Δf) and the estimated change (df). It is calculated as |Δf - df|. Understanding this error is important for evaluating the accuracy of linear approximations and the effectiveness of using derivatives to predict function behavior over small intervals.