When \(f \colon I \to \R\) is differentiable, we obtain a function \(f' \colon I \to \R\text{.}\) The function \(f'\) is called the first derivative of \(f\text{.}\) If \(f'\) is differentiable, we denote by \(f'' \colon I \to \R\) the derivative of \(f'\text{.}\) The function \(f''\) is called the second derivative of \(f\text{.}\) We similarly obtain \(f'''\text{,}\)\(f''''\text{,}\) and so on. With a larger number of derivatives the notation would get out of hand; we denote by \(f^{(n)}\) the \(n\)th derivative of \(f\text{.}\) When \(f\) possesses \(n\) derivatives, we say \(f\) is \(n\) times differentiable.
Named for the English mathematician Brook Taylor (1685–1731). It was found by the Scottish mathematician James Gregory (1638–1675). The statement we give is due to Joseph-Louis Lagrange (1736–1813).
(at least the version we give here) is a generalization of the mean value theorem. Mean value theorem says that up to a small error \(f(x)\) for \(x\) near \(x_0\) can be approximated by \(f(x_0)\text{:}\)
where the “error” \(f'(c)(x-x_0)\) is measured in terms of the first derivative at some point \(c\) between \(x\) and \(x_0\text{.}\) Taylor’s theorem generalizes this result to higher derivatives. It tells us that up to a small error, any \(n\) times differentiable function can be approximated at a point \(x_0\) by a polynomial of degree \(n\text{.}\) The error of this approximation goes to zero faster than \({(x-x_0)}^{n}\) as \(x\) goes to \(x_0\text{.}\) To see why this is a good approximation, notice that for a big \(n\text{,}\)\({(x-x_0)}^n\) is very small in a small interval around \(x_0\text{.}\) We will require one more (so \(n+1\)) derivative to write the error as we do in the mean value theorem.
For an \(n\) times differentiable function \(f\) defined near a point \(x_0 \in \R\text{,}\) define the \(n\)th order Taylor polynomial for \(f\) at \(x_0\) as
See Figure 4.8 for the odd-degree Taylor polynomials for the sine function at \(x_0=0\text{.}\) The even-degree terms are all zero, as even derivatives of sine are again sines, which are zero at the origin.
Suppose \(f \colon [a,b] \to \R\) is a function with \(n\) continuous derivatives on \([a,b]\) and such that \(f^{(n+1)}\) exists on \((a,b)\text{.}\) Given distinct points \(x_0\) and \(x\) in \([a,b]\text{,}\) there is a point \(c\) between \(x_0\) and \(x\) such that
The term \(R_n^{x_0}(x)\coloneqq
f(x)-P_n^{x_0}(x) =
\frac{f^{(n+1)}(c)}{(n+1)!}{(x-x_0)}^{n+1}\) is called the remainder term. This form is called the Lagrange form of the remainder. There are other ways to write the remainder term, but we skip those. Note that \(c\) depends on both \(x\) and \(x_0\text{.}\)
We compute the \(k\)th derivative at \(x_0\) of the Taylor polynomial \({(P_n^{x_0})}^{(k)}(x_0) = f^{(k)}(x_0)\) for \(k=0,1,2,\ldots,n\) (the zeroth derivative of a function is the function itself). Therefore,
In particular, \(g(x_0) = 0\text{.}\) We also have \(g(x) = 0\text{.}\) By the mean value theorem, there exists an \(x_1\) between \(x_0\) and \(x\) such that \(g'(x_1) = 0\text{.}\) Applying the mean value theorem to \(g'\text{,}\) we obtain that there exists \(x_2\) between \(x_0\) and \(x_1\) (and therefore between \(x_0\) and \(x\)) such that \(g''(x_2) = 0\text{.}\) We repeat the argument \(n+1\) times to obtain a number \(x_{n+1}\) between \(x_0\) and \(x_n\) (and therefore between \(x_0\) and \(x\)) such that \(g^{(n+1)}(x_{n+1}) = 0\text{.}\)
In the proof, we found \({(P_n^{x_0})}^{(k)}(x_0) = f^{(k)}(x_0)\) for \(k=0,1,2,\ldots,n\text{.}\) Therefore, the Taylor polynomial has the same derivatives as \(f\) at \(x_0\) up to the \(n\)th derivative. That is why the Taylor polynomial is a good approximation to \(f\text{.}\) Notice how in Figure 4.8 the Taylor polynomials are reasonably good approximations to the sine near \(x=0\text{.}\)
We do not necessarily get good approximations by the Taylor polynomial everywhere. Consider expanding the function \(f(x) \coloneqq \frac{x}{1-x}\) around 0, for \(x < 1\text{,}\) we get the graphs in Figure 4.9. The dotted lines are the first, second, and third degree approximations. The dashed line is the 20th degree polynomial. The approximation does seem to get better as the degree rises for \(x > -1\text{.}\) For \(x < -1\text{,}\) it in fact gets visibly worse. The polynomials are the partial sums of the geometric series \(\sum_{n=1}^\infty x^n\text{,}\) and the series only converges on \((-1,1)\text{.}\) See the discussion of power series in Section 2.6.
Figure4.9.The function \(\frac{x}{1-x}\text{,}\) and the Taylor polynomials \(P_1^0\text{,}\)\(P_2^0\text{,}\)\(P_3^0\) (all dotted), and the polynomial \(P_{20}^0\) (dashed).
There is no guarantee that this series converges for any \(x \neq x_0\text{.}\) Even where it does converge, there is no guarantee that it converges to the function \(f\text{.}\) Functions \(f\) whose Taylor series at every point \(x_0\) converges to \(f\) in some open interval containing \(x_0\) are called analytic functions. Many functions one tends to see in practice are analytic. See Exercise 5.4.11, for an example of a non-analytic function.
The definition of derivative says that a function is differentiable if it is locally approximated by a line. We mention in passing that there exists a converse to Taylor’s theorem, which we will neither state nor prove, saying that if a function is locally approximated in a certain way by a polynomial of degree \(d\text{,}\) then it has \(d\) derivatives.
Taylor’s theorem gives a quick proof of a version of the second derivative test. By a strict relative minimum of \(f\) at \(c\text{,}\) we mean that there exists a \(\delta > 0\) such that \(f(x) > f(c)\) for all \(x \in (c-\delta,c+\delta)\) where \(x\neq c\text{.}\) A strict relative maximum is defined similarly. Continuity of the second derivative is not needed, but the proof is more difficult and is left as an exercise. The proof also generalizes into the \(n\)th derivative test, which is also left as an exercise.
Suppose \(f \colon (a,b) \to \R\) is twice continuously differentiable, \(x_0 \in (a,b)\text{,}\)\(f'(x_0) = 0\) and \(f''(x_0) > 0\text{.}\) Then \(f\) has a strict relative minimum at \(x_0\text{.}\)
As \(f''\) is continuous, there exists a \(\delta > 0\) such that \(f''(c) > 0\) for all \(c \in (x_0-\delta,x_0+\delta)\text{,}\) see Exercise 3.2.11. Take \(x \in (x_0-\delta,x_0+\delta)\text{,}\)\(x \neq x_0\text{.}\) Taylor’s theorem says that for some \(c\) between \(x_0\) and \(x\text{,}\)
Suppose \(p\) is a polynomial of degree \(d\text{.}\) Given \(x_0 \in \R\text{,}\) show that the \(d\)th Taylor polynomial for \(p\) at \(x_0\) is equal to \(p\text{.}\)
Suppose \(f \colon \R \to \R\) has \(n\) continuous derivatives. Show that for every \(x_0 \in \R\text{,}\) there exist polynomials \(P\) and \(Q\) of degree \(n\) and an \(\epsilon > 0\) such that \(P(x) \leq f(x) \leq Q(x)\) for all \(x \in
[x_0,x_0+\epsilon]\) and \(Q(x)-P(x) = \lambda {(x-x_0)}^n\) for some \(\lambda \geq 0\text{.}\)
Suppose \(f \colon [a,b] \to \R\) has \(n+1\) continuous derivatives and \(x_0 \in (a,b)\text{.}\) Prove: \(f^{(k)}(x_0) = 0\) for all \(k = 0, 1, 2, \ldots, n\) if and only if \(\lim\limits_{x\to x_0} \frac{f(x)}{{(x-x_0)}^{n+1}}\) exists.
Suppose \(a,b,c \in \R\) and \(f \colon \R \to \R\) is twice differentiable, \(f''(x) = a\) for all \(x\text{,}\)\(f'(0) = b\text{,}\) and \(f(0) = c\text{.}\) Find \(f\) and prove that it is the unique differentiable function with this property.
(Challenging) Show that a simple converse to Taylor’s theorem does not hold. Find a function \(f \colon \R \to \R\) with no second derivative at \(x=0\) such that \(\babs{f(x)} \leq \babs{x^3}\text{,}\) that is, \(f\) goes to zero at 0 faster than \(x^2\text{,}\) and while \(f'(0)\) exists, \(f''(0)\) does not.
Show that there exists a once differentiable function \(g \colon [0,1) \to \R\) such that \(f(x) = g(x)\) for all \(x \neq 0\text{.}\) Hint: See Exercise 4.2.14.
Prove the \(n\)th derivative test. Suppose \(n \in \N\text{,}\)\(x_0 \in (a,b)\text{,}\) and \(f \colon (a,b) \to \R\) is \(n\) times continuously differentiable, with \(f^{(k)}(x_0) = 0\) for \(k=1,2,\ldots,n-1\text{,}\) and \(f^{(n)}(x_0)
\neq 0\text{.}\) Prove:
If \(n\) is even, then \(f\) has a strict relative minimum at \(x_0\) if \(f^{(n)}(x_0) > 0\) and a strict relative maximum at \(x_0\) if \(f^{(n)}(x_0) < 0\text{.}\)
Prove the more general version of the second derivative test. Suppose \(f \colon (a,b) \to \R\) is differentiable and \(x_0 \in (a,b)\) is such that, \(f'(x_0) = 0\text{,}\)\(f''(x_0)\) exists, and \(f''(x_0) > 0\text{.}\) Prove that \(f\) has a strict relative minimum at \(x_0\text{.}\) Hint: Consider the limit definition of \(f''(x_0)\text{.}\)
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