Calculus Small Changes, Big Ideas

Section 2.4 Linear approximation

• Thinking of the derivative as the ‘best linear approximation’ to a function at a point

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• Introduce the tangent line

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• Approximation using differentials

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• Linearization of a function at \(x=a\text{:}\)

  \begin{equation*} L(x)=f(a)+f'(a)(x-a) \end{equation*}

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• Using this idea when you only have data rather than a symbolic formula for the function

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• Use second derivative for over/underestimate analysis

  • If \(f''(a)>0\text{,}\) then \(f\) is concave up around \(a\text{,}\) so \(L(x)\lt L(a)\text{.}\)

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  • If \(f''(a)\lt 0\text{,}\) then \(f\) is concave down around \(a\text{,}\) so \(L(x)>L(a)\text{.}\)

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• Absolute error: \(\eps=|y-y\tsub{approx}|\)

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• Relative error: \(\eta=\dfrac{\eps}{|y|}\)

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• Percent error: \(\eta\tsub{percent}=\eta\times 100\%\)

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