Linear Approximation

LINEAR APPROXIMATION

The derivative supports a quick and effective means for approximating the values of complicated functions. The technique is known as linear approximation and it is really quite simple.

A differentiable function is one for which there is a tangent line at each point on the graph. In an intuitive sense, the tangent to a curve at a point is the line that looks most like the curve at the point of tangency. Assuming that f is differentiable at a, the tangent line to the graph y = f(x) at the point (a,f(a)) is given by the equation.

y - f(a) = f '(a)(x - a)
This equation arises from the point-slope formula for the line passing through (a,f(a)) with slope f '(a). The tangent line is also the graph of a function; we call this function L_a, for "linear." The graph of L_a is the tangent line to y=f(x) at (a,f(a)). Since the tangent line looks more like the graph than any other line (at least near (a,f(a))), the function L_a is the best linear approximation to f near a.

L_a(x) = f(a) + f '(a)(x - a)

The linear function L_a(x) is an approximation to f(x) in the following sense.

If x is close to a, then L_a(x) is close to f(x)

f(x)

L_a(x) = f(a) + f '(a)(x - a)

Example. Take f(x) = sin(x) and a=0. To construct the best linear approximation of sin(x) near 0, we compute f(a) and f '(a).

f(x) = sin(x); f(a) = f(0) = sin(0) = 0

f '(x) = cos(x); f '(a) = f '(0) = cos(0) = 1

With a=0, we find

L₀(x) = f(0) + f '(0)(x-0) = x.

Of course, the graph of the linear function L₀(x)=x is simply the tangent to the graph of f(x)=sin(x) at (0,f(0))=(0,0). Our focus here is on the fact that for values of x close to 0, f(x) is very close to L₀(x). We also encountered this fact when we studied the limit of sin(x)/x as x approaches 0.

Equipment Checks.

Find the best linear approximation to the function f(x) = 1/x near 2.

Find the best linear approximation to the function f(x) = 1/x near 5.

Find the best linear approximation to the function f(x) = sqrt(x) near 100.

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