Linear approximation: Difference between revisions

Revision as of 14:33, 18 April 2020

Revision as of 14:36, 18 April 2020

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:<math> f(x) \approx f(a) + f'(a)(x - a)</math>.

This is a good approximation for <math>f(x)</math> when <math>x</math> it is close enough to <math>a</math>; since a curve, when closely observed, will begin to resemble a straight line. Therefore, the expression on the right-hand side is just the equation for the [[tangent line]] to the graph of <math>f</math> at <math>(a,f(a))</math>. For this reason, this process is also called the '''tangent line approximation'''.

If <math>f</math> is [[concave down]] in the interval between <math>x</math> and <math>a</math>, the approximation will be an overestimate (since the derivative is decreasing in that interval). If <math>f</math> is [[concave up]], the approximation will be an underestimate.<ref>{{cite web|title=12.1 Estimating a Function Value Using the Linear Approximation|url=http://math.mit.edu/classes/18.013A/HTML/chapter12/section01.html|accessdate=3 June 2012}}</ref>