Section 2.9 Skills: Linear Approximations and Differentials

Section 2.9: Linear Approximations and Differentials

This section covers using derivatives to approximate function values and estimate errors.

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Skills in This Section

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Key Concepts

Linear Approximation

Near $x = a$, a differentiable function can be approximated by its tangent line:

$$f(x) \approx L(x) = f(a) + f'(a)(x - a)$$

Differentials

If $y = f(x)$, then:

• $dx$ is an independent variable (a small change in $x$)
• $dy = f'(x)\,dx$ is the corresponding change in $y$ along the tangent line

Error Estimation

For a measurement $x$ with error $dx$:

• Absolute error: $dy = f'(x)\,dx$
• Relative error: $\frac{dy}{y} = \frac{f'(x)}{f(x)}\,dx$

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