Taylor's Expansion for Function of Two Variables

TAYLOR'S EXPANSION FOR FUNCTION OF TWO VARIABLES

The Taylor's series expansion of a single variable function f(x) in a neighbourhood of a point a is

which is an infinite power series in h.

Maclaurin's series is

These ideas are extended to a function f(x, y) of two independent variables x, y. We state the theorem.

Theorem 3.4.1 Taylor's theorem

Let f(x, y) be a function of two independent variables x, y defined in a region R of the xy-plane and let (a, b) be a point in R. Suppose f(x, y) has all its partial derivatives in a neighbourhood of (a, b) then

Modified forms

1. Put x = a + h, y = b + k, then h = x - a, k = y - b

⸫ the Taylor's series can be written as

This series is known as the Taylor's expansion of f(x, y) in the neighbourhood of (a, b) or about the point (a, b).

2. Putting a = 0, b = 0 we get the expansion of f(x, y) in the neighbourhood of (0, 0)

This is called Maclaurin's series for f(x, y) in powers of x and y.

Note Taylor's formula gives polynomial approximation to a function of two variables about a given point.

WORKED EXAMPLES

Example 1 

Expand about (1, 1) upto the second degree terms.

Solution 

We know the expression of f(x, y) about the point (a, b) as Taylor's series is

Here (a, b) = (1, 1)

Example 2 

Expand sin (xy) in powers of (x - 1) and upto the second degree terms.

Solution 

We know Taylor's expansion about the point (a, b) is

we have f(x, y) = sin(xy)

At the point (1, π/2)

Example 3 

Expand ex cos y near the point (1, π/4) by Taylor's series as far as quadratic terms.

Solution 

We know Taylor's series about the point (a, b) is

Example 4 

Expand ex loge (1 + y) in powers of x and y upto terms of third degree. 

Solution 

Required the expansion in powers of x and y and so Maclaurin's series is to be used.

We know f(x, y) = ƒ(0,0) + [x ƒx (0, 0) + y ƒy(0, 0)]

Given f(x, y) = ex log (1 + y)

At the point (0, 0)

Example 5 

Expand x2 y + 3y - 2 in powers of x - 1 and y + 2 using Taylor's theorem.

Solution 

We know

Here (a, b) = (1, −2)

Note Since the given function is 3rd degree in x, y, the expansion terminates. 

Example 6 

If f(x, y) = tan-1(xy) compute an approximate value of f(0.9, -1.2)

Solution 

We shall use Taylor's series to find the approximate value. The point (0.9, −1.2) is close to the point (1, −1). So we shall find the Taylor's series about (1, -1).

Here (a, b) = (1, −1)

Put x =0.9, y = -1.2

Note We have approximated tan-1 xy by a second degree polynomial in x and y. Using this polynomial we have found the approximate value of f(0.9, -1.2) = -0.8229

But by direct computation f(0.9, −1.2) = tan-1(−1.08)

= tan-1 1.08

= -0.8238

correctly upto 4 decimal places.

The error is only 0.0009, which is negligible.

Example 7 

Find the Taylor series expansion of ex sin y at the point upto 3rd degree terms.

Solution 

Let f(x, y) = ex sin y

We know Taylor series expansion of f(x, y) about (a, b) is

Example 8 

Expand ex cos y in powers of x and y at (0, 0) upto third degree term, by Taylor's theorem.

Solution 

Let f(x, y) = ex cos y

We know that Taylor's series about (a, b) is

Here (a, b) = (0, 0)

Taylor's series about (0, 0) is

Example 9 

Expand exy in powers of (x − 1) and (y - 1) upto third degree terms, by Taylor's series.

Solution 

Let f(x, y) = exy

The Taylor's series for f(x, y) about (a, b) is

Here (a, b) = (1, 1)

⸫ the Taylor's series expansion for exy about (1, 1) is

Example 10 

Obtain the Taylor series of x3 + y3 + xy2 in powers of (x − 1) (y − 2)

Solution 

Let f(x, y) = x3 + y3 + xy2

The Taylor series about (a, b) is

Hence (a, b) = (1, 2)

⸫ Taylor's series is

Example 11 

Expand ex sin(y) in powers of x and y up the third degree terms.

Solution 

Let f(x, y) = ex sin y

We know the Taylor's series about (0, 0) is

Here the point is (0, 0)

We have

⸫ the Taylor's series is

EXERCISE 3.4

ANSWERS TO EXERCISE