What Are the Mean Value and Taylor Theorems Saying?

We have studied two propositions about the derivative of a function that sound vaguely alike.

(1) On the one hand, the mean value theorem (Week 13, Stewart 3.2) says that

f(x) = f(a) + f′(c) (x − a) (exactly!)

for some c between a and x. For example, e^{x} = 1 + e^{c}x.

(2) On the other hand, the best linear approximation (Week 16, Stewart 2.9) says that

f(x) = f(a) + f′(a) (x − a) + something small

if x is close to a.

For example, e^{x} ≈ 1 + x (that is, e^{x} = 1 + x + a small error). Quick exercise: What are the two crucial differences between (1) and (2)? Read closely.

An immediately interesting question about statement (2) is: How small? The word “small” by itself is too vague to be useful in technical work.

We can cast some light on the relation between (1) and (2) by stepping back a bit to the definition of continuity, which can be paraphrased this way:

(0) If f is continuous, then

f(x) = f(a) + something small

if x is close to a.

For example, e^{x} → e^{0} = 1 as x → 0. Again one should ask, “How small?” In this case the answer is given by the detailed definition of continuity (through the definition of a limit): For every ϵ there is a δ such that if |x − a| < δ, then |f(x) − f(a)| < ϵ. We will talk about statements like this in more depth in Week 26; for now the only thing to understand is that (0) merely says that as x − a goes to 0, f(x) − f(a) also goes to 0 — it says nothing about how fast f(x) − f(a) approaches 0.

But now look again at the mean value theorem (alias “Taylor’s theorem with N = 0”): It says that if f is differentiable, then

f(x) = f(a) + f′(c) (x − a)

and so, if |x − a| < δ, then*

|f(x) − f(a)| < δ max_{c∈(a,x)} |f′(c)|.

* c ∈ (a, x) means a < c < x, which is the right formulation if a < x; if x < a, of course, we would have c ∈ (x, a).

The important thing about this inequality is that it says that the difference between f(x) and f(a) is (at worst) linear in δ, the distance between x and a. Thus (1) is a sharpened form of (0), valid if f is differentiable, not just continuous.

But differentiability also gives us statement (2), and we want to know what “small” means in it. One type of answer is provided by Taylor’s theorem with N = 1:

(3) If the second derivative of f exists, then

f(x) = f(a) + f′(a) (x − a) + \frac{1}{2}f′′(c) (x − a)^{2}

for some c.

For example, e^{x} = 1 + x + \frac{1}{2}e^{c}x^{2}. Therefore, if |x − a| < δ, then

|f(x) − [f(a) + f′(a) (x − a)]| < \frac{1}{2}δ^{2} max |f′′(c)|.

That is, the error is quadratically small in its dependence on δ.

If f′(a) exists but f′′(a) doesn’t, then it turns out that the error in the linear approximation (2) typically goes to zero faster than δ but not as fast as δ^{2}. (Contemplate, for example, f(x) = x^{4/3} around x = 0.) We shall not discuss this sort of situation further (leaving it to advanced courses in mathematics).

Of course, when f′′ exists we expect to form a best quadratic approximation:

(4) If x is close to a, then

f(x) ≈ f(a) + f′(a) (x − a) + \frac{1}{2}f′′(a) (x − a)^{2}.

For example, e^{x} ≈ 1 + x + \frac{1}{2}x^{2}.

We can ask the usual question: Exactly how small is the error in the approximation (4)? By now you can probably guess the answer:

(5) If the third derivative exists, then (Taylor’s theorem with N = 2)

f(x) = f(a) + f′(a) (x − a) + \frac{1}{2}f′′(a) (x − a)^{2} + \frac{1}{6}f^{(3)}(c) (x − a)^{3}

for some c, and therefore the error in the quadratic approximation is cubic in δ.

Obviously we can continue this game forever; successfully completing each step invites us to attempt the next step. If f^{(N)}(a) exists, then we can construct the Nth Taylor polynomial

T_{N}(x) = ∑_{j=0}^{N} \frac{f^{(j)}(a)}{j!} (x − a)^{j}

and expect it to be the best approximation (near a) to f(x) by a polynomial of degree N. (This is proposition number (2N).) We ask how good this approximation really is. Then Taylor’s theorem says that if f^{(N+1)}(a) exists, then

f(x) = T_{N}(x) + R_{N}(x),

with

R_{N}(x) = \frac{f^{(N+1)}(c)}{(N + 1)!} (x − a)^{N+1},

and therefore the error in the Nth-degree approximation vanishes as fast as δ^{N+1} as δ = |x − a| approaches 0. (This is proposition number (2N +1).) This formula tempts us to consider the even better approximation T_{N+1} , but we have to stop somewhere.