Math 2.0

Real functions Concept

Linear approximation

Builds on Total derivativeTangent plane to a graph

Summary

A linear approximation of a function ff near a point aa replaces ff by an affine function. This function has the same value and the same derivative at aa. For a differentiable map ff from an open subset of ℝn\mathbb{R}^n to ℝm\mathbb{R}^m, it is f(x)≈f(a)+Df(a)(x−a)f(x) \approx f(a) + Df(a)(x - a), where Df(a)Df(a) is the total derivative. The error is small compared with ∥x−a∥\|x - a\| as x→ax \to a.

Remarks

For a function of one variable, the graph of the linear approximation is the tangent line. For a function of two variables, it is the tangent plane. For f(x)=xf(x) = \sqrt{x} near a=4a = 4, the approximation x≈2+(x−4)/4\sqrt{x} \approx 2 + (x - 4)/4 gives 4.1≈2.025\sqrt{4.1} \approx 2.025, and the true value is 2.0248…2.0248\ldots.

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To understand this page from the ground up, read these 8 pages in order. The foundations come first.

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Euclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptGraph of a function, concept, no evidence levelGraph of a functionconceptTotal derivative, concept, no evidence levelTotal derivativeconceptFunction of several variables, concept, no evidence levelFunction of several v…conceptDifferentiable function of several variables, concept, no evidence levelDifferentiable functi…conceptTangent plane to a graph, concept, no evidence levelTangent plane to a gr…conceptLinear approximation, concept, no evidence levelLinear approximationconcept
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Euclidean space object
  2. Function concept
  3. Graph of a function concept
  4. Total derivative concept
  5. Function of several variables concept
  6. Differentiable function of several variables concept
  7. Tangent plane to a graph concept
  8. Linear approximation concept

Further reading

  1. Web https://en.wikipedia.org/wiki/Linear_approximation, Definition
About this page
Tag0W276SSSCG
URIm2://0W276SSSCG
Aliaseslinearisation, tangent-line-approximation
Kindconcept
WritingW0 stub
MSC26B05
UsesTotal derivative, Tangent plane to a graph
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