Taylor Polynomials

Skip to main content

Section 4.7 Taylor Polynomials

If

g

is a function of one variable then the Taylor polynomial of order

n

about

a

is defined by

T_{n} (a + h) = g (a) + g^{'} (a) h + \frac{g^{″} (a)}{2!} h^{2} + \dots + \frac{g^{(n)} (a)}{n!} h^{n} .

As you know from first year calculus

T_{n}

is the best approximation of

f

near

a

by an

n

-th order polynomial. From the above formula we now want to derive the Taylor polynomials of order one and two for a function,

f,

defined on

D \subset R^{N} .

Given an interior point of

a \in D

and

h \in R^{N}

we consider the function

g_{h} (t) := f (a + t h),

and write down its Taylor polynomial of order

2

about

t = 0 :

g_{h} (0) + g_{h}^{'} (0) t + \frac{g_{h}^{″} (0)}{2!} t^{2} .

Applying the chain rule (see Theorem 4.17) we obtain

g_{h}^{'} (0) = (grad f (a)) \cdot h .

If

h \neq 0

then

v := h / ‖ h ‖

is a unit vector, and by the formula (4.3) the above becomes

g_{h}^{'} (0) = ‖ h ‖ (grad f (a)) \cdot v = ‖ h ‖ \frac{\partial f}{\partial v} (a) .

Applying the same arguments to

g_{h}^{'} (t)

we get

g_{h}^{″} (0) = ‖ h ‖^{2} \frac{\partial^{2} f}{\partial v^{2}} (a) .

Using the formula for the second directional derivative from Proposition 4.43 and the definition of

v

we get

\begin{aligned} g_{h}^{″} (0) & = ‖ h ‖^{2} v^{T} H_{f} (a) v \\ = ‖ h ‖^{2} \frac{h^{T}}{‖ h ‖} H_{f} (a) \frac{h}{‖ h ‖} \\ = h^{T} H_{f} (a) h, \end{aligned}

where

H_{f} (a)

is the Hessian matrix as defined in Definition 4.40. The above motivates the following definition.

Definition 4.44. Taylor polynomials.

The functions

\begin{aligned} T_{1} (a + h) & := f (a) + (grad f (a)) \cdot h \\ T_{2} (a + h) & := f (a) + (grad f (a)) \cdot h + \frac{1}{2} h^{T} H_{f} (a) h \end{aligned}

are called the Taylor polynomial of

f

of order one and two, respectively, centred at

a .

By construction

T_{1}

and

T_{2}

are the Taylor polynomials of

g_{h}

at

t = 1,

so we can apply Taylor’s theorem to

g_{h}

at

t = 1 .

Doing this, with some effort involved in dealing with the remainder term, we get the following version of Taylor’s Theorem for a function of several variables. For a rigorous proof we refer to Apostol [2](pages 308-310).

Theorem 4.45. Taylor’s Theorem.

Suppose that all first and second order partial derivatives of

f

are continuous at

a .

Then for

n = 1, 2

f (a + h) = T_{n} (h) + R_{n} (h),

where the remainder term

R_{n} (h)

has the property that

lim_{h \to 0} \frac{R_{n} (h)}{‖ h ‖^{n}} = 0,

where

n = 1, 2

and

T_{n}

are as in Definition 4.44.

We conclude this section by giving one example.

Example 4.46.

Suppose that

f (x, y) := x^{3} - 3 x^{2} y

for all

(x, y) \in R^{2}

is as in Example 4.37. Then we saw that

grad f (x, y) = (3 x^{2} - 6 x y, - 3 x^{2})

and

H_{f} (x, y) = [\begin{matrix} 6 x - 6 y & - 6 x \\ - 6 x & 0 \end{matrix}] .

Hence the Taylor polynomial of order one centred at

a = (1, 1)

is

\begin{aligned} T_{1} (1 + h_{1}, 1 + h_{2}) & = - 2 + (- 3, - 3) \cdot (h_{1}, h_{2}) \\ = - 2 - 3 (h_{1} + h_{2}) \end{aligned}

for all

(h_{1}, h_{2}) \in R^{2} .

The Taylor polynomial of order two centred at

a = (1, 1)

is

\begin{aligned} T_{2} (1 + h_{1}, 1 + h_{2}) & = T_{1} (1 + h_{1}, 1 + h_{2}) + \frac{1}{2} [\begin{array}{c} h_{1} & h_{2} \end{array}] [\begin{array}{c} 0 & - 6 \\ - 6 & 0 \end{array}] [\begin{array}{c} h_{1} \\ h_{2} \end{array}] \\ = - 2 - 3 (h_{1} + h_{2}) - 6 h_{1} h_{2} . \end{aligned}

web analytics