Numerical Methods 3

Taylor series

Denny Vitasari

10 March 2015

Denny Vitasari Taylor series

Case study

Figure: Tacoma Narrows Bridge, 1 July 7 November 1940


Case Study

The original Tacoma Narrow Bridge in Washington, US was openedon 1 July 1940 and collapsed in the morning of 7 November 1940under high wind conditions.

At the time it was the third longest bridge in the world.The collapse was caused by torsional twisting from side-to-side.Engineers, architects, and physicists attempted to determinethe cause of the collapse.One of the model considered two dimensional slides, ignoringthe dimensional length of the bridge, interested only on theside-to-side motion.This approximation needs a knowledge on local behaviours ofthe function involved.


Taylor polynomial

Taylor polynomials give a convenient way to describe the localbehaviour of a function, by encapsulating its first several derivativesat a point.

Good approximation needs a good knowledge on the error.Error: Taylor polynomial vs the exact value.Question on Taylor theorem: to what extent do the derivativesof a function at a single point dictate the behaviour of thefunction at nearby points?


Taylor series are used to estimate the value of functions (at leasttheoretically - nowadays we can usually use a calculator orcomputer to calculate directly.)

Figure: f (x) = x2sin(x/2) and tangent line.


The mean value theorem

The derivative of a function at a point is the slope of that functionon that point.

Figure: f (x) = x2sin(x/2) and f (x) at x = 2


The slope of a line between a and b on the graph of y = f (x) isequal to the derivative of f (x) somewhere between the two points.

In other words, take c as a point between a and b:

f (c) =f (b) f (a)

b a


Rearrange the equation:

f (b) f (a) = f (c)(b a)


Taylor theorem

If f (x) is differentiable on an open interval I containing a then forany x in I :

f (x) = f (a) + (x a)f (c)for some c between a and x . If x is close to a, then c is also closeto a, and we can approximate:

f (x) f (a) + f (a)(x a)

In other words, near x = a the function f (x) is well-approximatedby the LINEAR function L(x) = f (a) + f (a)(x a) which is atangent line approximation.


If f (x) is twice differentiable on an open interval I containing athen for any x in I :

f (x) = f (a) + f (a)(x a) + 12f (c)(x a)2

for some c between a and x .If x is close to a, then c is also close to a:

f (x) f (a) + f (a)(x a) + 12f (a)(x a)2


Linear approximation:

f (x) f (a) + f (a)(x a)

Quadratic approximation:

f (x) f (a) + f (a)(x a) + 12f (a)(x a)2


Taylor series expansion

The Taylor expansion of f (x) around a:

f (x) f (a)+f (a)(xa)+12f (2)(a)(xa)2+ 1

2 3 f(3)(a)(xa)3+...

Or in other format:

f (x) k=0

1k!f (k)(a)(x a)k


Example: Taylor expansion of y = sin x around x = 0


Maclaurin series

Maclaurin series is a special case of Taylor series with the centre ofexpansion is a = 0.The Maclaurin expansion of f (x):

f (x) f (0) + f (0)x + 12f (2)(0)x2 +

12 3 f

(3)(0)x3 + ...

Or in other format:

f (x) k=0

1k!f (k)(0)xk


Taylor and Maclaurin series: Facts

The Taylor series is named after the English mathematicianBrook Taylor (16851731).The Maclaurin series is named for the Scottish mathematicianColin Maclaurin (16981746).Maclaurin series are named after Colin Maclaurin because hepopularized them in his calculus textbook Treatise of Fluxionspublished in 1742.


Example:Maclaurin series for f (x) = sin xThe derivatives:

f (x) = cos xf (x) = sin xf (3)(x) = cos xf (4)(x) = sin xf (5)(x) = cos x

f (x) = 1f (x) = 0f (3)(x) = 1f (4)(x) = 0f (5)(x) = 1

Therefore we obtain:

sin x x x3

3!+

x5

5! x

7

7!+ ...


Exercise

Find the Maclaurin series of:f (x) = cos xf (x) = ex

f (x) = (1 x)1


Remainder

A Taylor series:

f (x) k=0

1k!f (k)(a)(x a)k

Taylor polynomial of order n:

Pn(x) n

k=0

1k!f (k)(a)(x a)k

Remainder:

Rn(x)

k=n+1

1k!f (k)(a)(x a)k


When we use part of a Taylor series to estimate the value of afunction, the end of the series that we do not use is called theremainder. If we know the size of the remainder, then we know howclose our estimate is.


An amazing use of infinite series

ex = 1+ x +x2

2!+

x3

3!+

x4

4!+ ...

Substitute xi for x

e ix = 1+ ix +(ix)2

2!+

(ix)3

3!+

(ix)4

4!+

(ix)5

5!+

(ix)6

6!+ ...

e ix = 1+ ix +i2x2

2!+

i3x3

3!+

i4x4

4!+

i5x5

5!+

i6x6

6!+ ...

e ix = 1+ ix x2

2! ix

3

3!+

x4

4!+

ix5

5! x

6

6!+ ...

e ix = 1 x2

2!+

x4

4! x

6

6!+ ...+ i

(x x

3

3!+

x5

5!+ ...

)


e ix = 1 x2

2!+

x4

4! x

6

6!+ ...+ i

(x x

3

3!+

x5

5!+ ...

)e ix = cos x + i sin x

let x = pie ipi = cospi + i sinpi

e ipi = 1+ i 0e ipi + 1 = 0

This is an amazing identity shows the interrelation of five mostfamous numbers in mathematics.


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Numerical Methods 3