CHAPTER 2 SECTION 9

August 1999

One of the most important applications of the derivative
is the construction of lines (or polynomials

of some fixed degree) that approximate a complex function near some point (we
call this

local approximation, and in the case of lines, linear approximation). In this
section, we develop

formulas for the construction of both approximating lines and parabolas for
functions.

The single most important thing to remember about the
derivative is that it tells us how best to

(locally) approximate a function by a straight line (the tangent line to the
graph). The slope of

this line is also referred to as the instantaneous rate of change of the
function (at the point under

consideration).

We have already discussed the fact that the tangent line
is the best local approximation to a

function. The equation of the tangent line to the graph of a function f at a
point (x_{0}, f(x_{0})) is:

y = f(x_{0}) + f'(x_{0}) (x - x_{0}) .

So, the idea is that, for x close to x_{0},

f(x) ≈ f(x_{0}) + f'(x_{0}) (x - x_{0}) .

This is called linear approximation; we are approximating a function with a line.

Let’s remind ourselves how this works by looking at an example. Suppose we
consider the

function

f(x) = sqrt(x).

Then

We know the exact value of both f and f' at any perfect
square (1, 4, 9, 16, ...), and can readily

find the equation of the tangent line to the graph at any point corresponding to
one of these.

Let’s look at the tangent line at the point (16,4), together with the original function

We see that the tangent line remains close to the graph of
the function (is a good approximation)

close to the point (16,4). We can use the simpler equation of the tangent line
to approximate the

function.

Here is a plot of the difference between the two:

Note that the approximation appears to be very good between 14 and 18:

Note that, at the point (x_{0}, f(x_{0})), the derivative of the
function that defines the tangent line has

the same value as the derivative of f at that point. That’s really obvious since
we construct the

tangent line with slope f' (x_{0}). The point is that the tangent line matches the
original funcion and

its derivative at the point (x_{0}, f(x_{0})).

We can extend this notion of local approximation to higher orders. The next step
is to consider

quadratic approximations.

For a function f, we wish to construct a quadratic polynomial which is the best
approximation

to f near a given point (x_{0}, f(x_{0})). Following the reasoning above, we want to
find a quadratic

function which passes through (x_{0}, f(x_{0})) and has the same first and second
derivatives as f at

that point. This is best done by writing a quadratic in the form:

q(x) = c + b (x − x_{0}) + a (x − x_{0})^{2} .

Remember that we want q and f to satisfy

q (x_{0}) = f (x_{0})

q'(x_{0}) = f'(x_{0})

q"(x_{0}) = f" (x_{0}) .

The first condition immediately yields c = f(x_{0}).

Since

q'(x) = b + 2a(x - x_{0}),

the condition q'(x_{0}) = f'(x_{0}) requires that we choose

b = f'(x_{0}) .

Note that the linear piece of our quadratic function is the same as the tangent
line. Finally, since

q"(x) = 2a, the condition q" (x_{0}) = f" (x_{0}) means that we must take

So, our quadratic approximation becomes

In our example above, we looked at the linear
approximation to the graph of f(x) = √x at the

point (16,4). Since

the quadratic approximation at the same point is

Let’s look at a plot of this, together with a plot of f, to see how close the approximation is.

This approximation looks very good near (16,4). Let’s look
at the plot of the difference over a

smaller interval.

Comparing this approximation to the linear approximation, it appears that we have improved the

accuracy by about a factor of 10. That’s pretty impressive, and fairly representative of what happens with other examples.