apcalcLinear+Approximation

= Linear Approximations =

Definition :
A curve lies very close to its tangent line near the point of tangency. At a very close observation, that part of the graphs looks more and more like its tangent line.This observation is the basis for finding approximate values of functions. In general, one uses the tangent line at (//a,f// (//a//)) as an approximation to the curve //y// = //f// (//x//) when x is near a.An equation of this tangent line is //y//=//f// (//a//) +//f�// (//a//) (//x-a//)and the approximation //f// (//x//) ≈ //f// (//a//) + //f�// (//a//) (//x-a//) is called the __linear approximation__ or __tangent line approximation__ of //f// of //a//. The linear function whose graph is this [|tangent line], //L//(//x//) = //f//(//a//) + //f�// (//a//) (//x-a//), is called the linearizationof //f// of //a//.

History :
The linear approximation formula arises from the definition of the [|derivative]of a function //y// //f// (//x//)at a point //x// //a//:

It follows that for x near a ,

If you multiply across by) and rearrange that, you get the linear approximation equation (where x is near a):

Example :
Consider the function. The tangent line to this function at is ( is the valueat which we are finding the tangent line.). So here, for x near x = 1,. Then approximate :

If you use a calculator, the answer to four decimal places is 1.0488, which is extremely close to our approximation.

However, if we try to use the same linear approximation for an x value far from x=1, the result is less accurate. Forexample,, which is further from the actual answer of 0.5.

For more help and examples, watch these videos: [] [] [] (this video can only be viewed if you are not on a school computer network)

Applications to Real Life :
Linear approximations are often used in physics. In analyzing the consequences of an equation, a physicist sometimes needs to simplify a function by replacing it with its linear approximation. For instance, in deriving a formula for the period of a [|pendulum], physics textbooks obtain the expression for a tangential acceleration and then replace by with the remark that is very close to if is not too large. You can verify that the linearization of the function at is and so the linear approximation at 0 is. So, in effect, the derivation of theformula for the period of a pendulum uses the tangent line approximation forthe sine function.

Another example occurs in the theory of [|optics], where light rays that arrive at shallow angles relative to the optical axis are called paraxial rays. In paraxialoptics, both and are replaced by their linear approximations.In other words, the linear approximations

and

are used because isclose to 0. The results of the calculations made with these approximations became the basic theoretical tool used to design lenses.

Works Cited :
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