Math Help

Linear Approximation of a Function

In a Nut Shell: The value of a function, f(x), may be easy to calculate at a given point, a,

but very difficult at a "neighboring" point, x, (close by point). A "linear approximation", L(x),

of the function provides a way to approximate f(x) at the neighboring point x.

The figure below depicts the linear approximation, L(x), of the function, f(x).

where f(x) is the value of the function f at x

f(a) is the value of the function f(x) at x = a;
pick x to be slightly larger or slightly smaller than a

x ˗ a is the interval; it is best to have a small interval

f '(a) is the slope of the function f(x) at x = a

L(x) is the linear approximation of f(x) near x = a; it is the value you are seeking.

Example: Use linear approximation to estimate the value of the square root of 9.1.

Here x = 9.1 and a = 1.0.

Strategy: First define the function involved. In this case f(x) is the square root function.

f(x) = x^1/2So f ' (x) = (1/2) x^-1/2

Since 9.1 is close to 9, the approximation may be reasonable to obtain f(9.1).

Apply the linear approximation: L(x) = f(a) + f'(a) (x ˗ a)

Here L(9.1) = f(9) + f '(9) (9.1 ˗ 9) = 3 + (1/6) (9.1 ˗ 9) = 3.0166666

The actual value of the square root of 9.1 is 3.0166206.

Note: Since 9.1 is fairly close to 9 you can expect a good approximation. If on the other

hand you were trying to estimate 9.5 you can expect a poorer approximation since the straight

line approximation of L(x) (figure above) will be further from the actual value, f(x).