The graph of the cosine function is a very nice looking curve. But it's just that, a curve. The best linear approximation to the cosine function near 0 is quite unexciting; you can check that for f(x)=cos(x), the best linear approximation near 0 is given by L0(x)=1.

 LINES ARE EASY!
One reason that linear approximations are popular is because linear functions are easy to work with. But as we see from the cosine function, linear approximations have their limitations.

 PARABOLAS ARE EASY TOO!
After linear functions, the next simplest functions are the quadratics. In college algebra, we learn a lot about how to deal with these functions. This suggests that it might be useful try to approxmate complicated curves with parabolas. A look at the gr aph of the cosine function near (0,1) might lead you to believe that this idea is a good one.

Suppose that we are given a function f(x) and a point a in the domain a of f. The best linear approximation to f(x) near a is given by the linear function

La(x) = f(a) + f '(a)(x-a)

The graph of the linear function La(x) passes through the point (a,f(a)) and has slope f '(a). The the functions f and La have the same value at a AND they have the same derivative at a.

La(a) = f(a)

L'a(a) = f '(a)

Of course, the second and higher derivatives of linear functions are all zero, but a quadratic function generally has two nonzero derivatives. So here's what we want: a quadratic function Qa(x) with the property that the functions f and Q a have the same value, first derivative, AND second dervative at a.

Qa(a) = f(a)

Q'a(a) = f '(a)

Q''a(a) = f ''(a)

Example. Consider the cosine function f(x)=cos(x) and let a=0. It's not hard to verify that if we let Q(x) = 1 - x2/2, then f and Q have the same first two derivatives at 0. In addition, the graph of Q resembles the graph of f very nice ly near 0.

 f(x) = cos(x) Q(x) = 1 - x2/2 f(0) = cos(0) = 1 Q(0) = 1 - 02 = 1 f '(x) = -sin(x) Q'(x) = -x f '(0) = -sin(0) = 0 Q'(0) = -0 = 0 f ''(x) = -cos(x) Q''(x) = -1 f ''(0) = -cos(0) = -1 Q'-(0) = -1

Equipment Check. Find a quadratic polynomial function Q(x) that has the same two first derivative values as the natural logarithm g(x)=ln(x) at x=1.

We leave it as a Practice exercise to confirm that the following quadratic function does the job for any given f and a. That is, with the quadratic Qa listed below, the functions Qa and f agree at a, as do the derivatives Q'a and f  ', as do the second derivatives Q''a and f ''

 Qa(x) = f(a) + f '(a)(x-a) + f ''(a)(x-a)2/2

When asked to produce a quadratic approximation , you can either use this formula or simply solve for the coefficients in the quadratic approximation directly as we did in solving the Equipment Check problem. This latter approach cuts down on the need to memorize another formula!