Linear approximation (or linearization) is a method of estimating a value on a function by placing that value on a tangent line of the function at a nearby value. This approximation can be found with the formula:


An upper bound for the error of this term is

with being the nearby value, being the exact value, and being the approximation. Since a tangent line is being used, accuracy decreases the further apart and are. A linear approximation is a Taylor approximation of the first degree.

Example

To find , the steps would be as follows:

(This is the operation being applied)

(The derivative of the function)

(All values substituted into the formula, is given the nearby value of )






This value is quite close to the actual square root of

which is approximately

.

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