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Applications of Derivatives · Applied Problems

Linearization

L(x)=f(a)+f′(a)(x−a)L(x) = f(a) + f'(a)(x - a)

The linearization of f at a is the tangent line, used as a linear approximation of f near a. This is the same as the first-degree Taylor polynomial.

  • 2 worked examples
  • 15 in Applications of Derivatives

Worked examples

Approximate √4.1 using linearization of f(x) = √x at a = 4.
  1. f(4) = 2, f'(x) = 1/(2√x), f'(4) = 1/4
  2. L(x) = 2 + (1/4)(x - 4)
  3. L(4.1) = 2 + (1/4)(0.1) = 2.025

Answer: √4.1 ≈ 2.025 (actual: 2.02485...)

1 more worked examplePremium

Approximate ∛8.06 using linearization of f(x) = x^(1/3) at a = 8.

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Common questions

What is the Linearization?

The linearization of f at a is the tangent line, used as a linear approximation of f near a. This is the same as the first-degree Taylor polynomial. It is one of the applications of derivatives formulas in the CalcRef reference.

How do you use the Linearization?

Worked example. Approximate √4.1 using linearization of f(x) = √x at a = 4. f(4) = 2, f'(x) = 1/(2√x), f'(4) = 1/4. L(x) = 2 + (1/4)(x - 4). L(4.1) = 2 + (1/4)(0.1) = 2.025. Answer: √4.1 ≈ 2.025 (actual: 2.02485...).

Practice

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