Linear approximation

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Tangent line at (a, f(a))
Tangent line at (a, f(a))

In mathematics, a linear approximation is an approximation of a general function using a linear function (more precisely, an affine function).

[edit] Definition

Given a differentiable function f of one real variable, Taylor's theorem for n=1 states that

f(x) = f(a) + f\ '(a)(x - a) + R_2

where R2 is the remainder term. The linear approximation is obtained by dropping the remainder:

f(x) \approx f(a) + f\ '(a)(x - a)

which is true for x close to a. The expression on the right-hand side is just the equation for the tangent line to the graph of f at (a, f(a)), and for this reason, this process is also called the tangent line approximation.

Linear approximations for vector functions of a vector variable are obtained in the same way, with the derivative at a point replaced by the Jacobian matrix. For example, given a differentiable function f(x,y) with real values, one can approximate f(x,y) for (x,y) close to (a,b) by the formula

f\left(x,y\right)\approx f\left(a,b\right)+\frac{\partial f}{\partial x}\left(a,b\right)\left(x-a\right)+\frac{\partial f}{\partial y}\left(a,b\right)\left(y-b\right).

The right-hand side is the equation of the plane tangent to the graph of z = f(x,y) at (a,b).

In the more general case of Banach spaces, one has

f(x) \approx f(a) + Df(a)(x - a)

where Df(a) is the Fréchet derivative of f at a.

[edit] Examples

To find an approximation of \sqrt[3]{25} one can do as follows.

  1. Consider the function f(x)= x^{1/3}.\, Hence, the problem is reduced to finding the value of f(25).
  2. We have
    f\ '(x)= 1/3x^{-2/3}.
  3. According to linear approximation
    f(25) \approx f(27) + f\ '(27)(25 - 27) = 3 - 2/27.
  4. The result, 2.926, lies fairly close to the actual value 2.924…

[edit] References

  • Weinstein, Alan; Marsden, Jerrold E. (1984). Calculus III. Berlin: Springer-Verlag, page 775. ISBN 0-387-90985-0. 
  • Strang, Gilbert (1991). Calculus. Wellesley College, page 94. ISBN 0-9614088-2-0. 
  • Bock, David; Hockett, Shirley O. (2005). How to Prepare for the AP Calculus. Hauppauge, NY: Barrons Educational Series, page 118. ISBN 0-7641-2382-3.