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|.


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f(x) = x² + 2x, x₀ = 1, dx = 0.1

Verified step by step guidance

1. First, calculate the change in the function, Δf = f(x₀ + dx) − f(x₀). Substitute x₀ = 1 and dx = 0.1 into the function f(x) = x² + 2x to find f(x₀ + dx) and f(x₀).
2. Evaluate f(x₀ + dx) by substituting x = x₀ + dx = 1 + 0.1 = 1.1 into the function: f(1.1) = (1.1)² + 2(1.1).
3. Evaluate f(x₀) by substituting x = x₀ = 1 into the function: f(1) = 1² + 2(1).
4. Calculate the derivative of the function, fʹ(x) = d/dx (x² + 2x). This gives fʹ(x) = 2x + 2. Then, find the estimate df = fʹ(x₀) dx by substituting x₀ = 1 and dx = 0.1.
5. Finally, compute the approximation error |Δf − df| by taking the absolute value of the difference between the change in the function Δf and the estimate df.

Key Concepts

Function Change Δf

The change in a function, Δf, represents the difference in the function's value as the input changes from x₀ to x₀ + dx. It is calculated by evaluating the function at the new point and subtracting the function's value at the original point: Δf = f(x₀ + dx) − f(x₀). This concept helps understand how the function behaves over small intervals.

Differential Estimate df

The differential estimate, df, is an approximation of the change in the function using the derivative at a specific point. It is calculated as df = fʹ(x₀) dx, where fʹ(x₀) is the derivative of the function at x₀, and dx is the small change in x. This linear approximation is useful for estimating function changes over small intervals.

Approximation Error |Δf − df|

The approximation error, |Δf − df|, measures the difference between the actual change in the function and the estimated change using the derivative. It quantifies the accuracy of the linear approximation provided by the differential. A smaller error indicates a better approximation, highlighting the importance of understanding the behavior of the function and its derivative.