]> 10.5 Accuracy of Approximations, and the Mean Value Theorem

10.5 Accuracy of Approximations, and the Mean Value Theorem

We now ask, how accurate are any of the approximations here, from the trivial one, the constant approximation f ( x ) = f ( x 0 ) , the linear approximation, etc.

Suppose x > x 0 , m is the minimum value of the k-th derivative of f between these two arguments, and M is the maximum value of that derivative there.

We will invoke a principle, which in its simplest form is the statement: the faster you move, the further you go, other things being equal. Here we claim that if we invent a new function f M by replacing the actual value of the k-th derivative of f throughout the interval ( x 0 , z ) by its maximum value over that interval, then f M and all its first k = 1 derivatives will obey f M ( x ' ) f ( j ) ( x ' ) for all x ' in that interval.

Think of it this way: if you increase your speed f ' to the value M you increase the distance traveled. If alternately you increase acceleration f " to M , by the same argument, that will increase speed, and hence will increase distance traveled. And so on. If you increase a higher derivative, that increase will trickle down to increase all the lower derivatives, and ultimately f itself.

The nice thing about doing this is that the degree k approximation to f M at x 0 is exact at argument x , because f M 's k-th derivative is constant in the interval between x 0 and x . Now the degree k approximation to f M is the degree k 1 approximation to f plus M ( x x 0 ) k k ! .

Our inequality above applied with j = 0 therefore tells us that the (k - 1)-th approximation to f, plus M ( x x 0 ) k k ! is at least f ( x ) , while by the same argument applied in the opposite order with M replaced by m , we can deduce that the same approximation plus m ( x x 0 ) k k ! is at most f ( x ) .

The upshot of all this is have bounds on how far off the degree k 1 approximation to f at x 0 is from f at argument x : their difference lies between m ( x x 0 ) k k ! and M ( x x 0 ) k k ! .

We can go one step further and notice that this tells us that the error in the degree k 1 approximation can be written as q ( x x 0 ) k k ! where q lies between m and M .

Since m and M are the minimum and maximum values of f ( k ) between x 0 and x , if f ( k ) takes on all values in between its maximum and minimum (which it must if it is differentiable in that interval), it will take on the value q . We can therefore write q = f ( k ) ( x ' ) for some x ' in that interval.

This allows us to translate our conclusion here into the following statement.

Theorem:

The error in the degree k 1 approximation to f at x 0 evaluated at argument x is

f ( k ) ( x ' ) ( x x 0 ) k k !

for some x ' in the interval ( x 0 , x ) , if f ( k ) is continuous in that interval.

Exercises:

10.5 State this theorem for k = 1 . This result is called "the mean value theorem".

10.6 Repeat the argument above for the situation that occurs when x < x 0 . How does the conclusion change? What is different in the argument?