FM Maclaurin Taylor series name _______________________

Objective Deadlines / Progress

Mac

laur

in S

erie

s exp

ansio

n

Understand the notation used and find particular nth derivatives and their values for given x

Derive the expansion of functions such as ex , ln x , sin x ,… from first principles

Find the series expansion of a function using Maclaurins expansion formula Use the result to approximate the value of the function for given x

Use the standard results to find series expansions of composite functions

Find the series expansion of a function where f(0) undefined using Taylors expansion formula

Adapt the Maclaurin series to find solutions of differential equations

FM Maclaurin Taylor series name _______________________

WB1 Given that y=ln (1−x ) Find the value of:

a¿ (d2 ydx2 )2 b¿(d3 ydx3 )12

FM Maclaurin Taylor series name _______________________

WB2 Given that f (x)=ex2

a) Show that f ' ( x )=2xf (x )b) By differentiating the result twice more with respect to x, find f’’(x) and f’’’(x)c) Deduce the values of f(0), f’(0), f’’(0) and f’’’(0)

WB3 Complete the table:

f (x) f ' (x) f ' ' (x) f ' ' ' (x)

e2x

(1+x)5

x ex

ln (1+ x)

FM Maclaurin Taylor series name _______________________

FM Maclaurin Taylor series name _______________________

WB4 Expansion the long way

Given that f(x) = ex can be written in the form:ex=a0+a1 x+a2 x

2+a3 x3+.. .+ar x

r

And that it is valid to differentiate an infinite series term by term, show that:

ex=1+x+ x2

2!+ x

3

3 !+…+ x

r

r ! Effectively, we need to find the values of the ‘a’ terms

https://www.geogebra.org/m/w1P4FBV6 demonstrates this (and other functions) graphically

FM Maclaurin Taylor series name _______________________

For the continuous function: f ( x ) , x∈RProvided that f(0), f’(0), f’’(0), …, fr(0) all have finite values, then:

f ( x )=f (0 )+ f ' (0 ) x+ f ' '(0)2 !

x2+ f ' ' ' (0)3!

x3…

You get given this result in the formula booklet!Some functions are only valid for certain values of x (similar to the binomial expansion)The ranges are given where appropriate in exam questions

WB5 Using Maclaurin series

Express ln (1+x)as an infinite series in ascending powers of x, up to and including the term in x3

Using this series, find approximate values for:a) ln(1.05) b) ln(1.25) c) ln(1.8)

FM Maclaurin Taylor series name _______________________

WB6 Express sin x as an infinite series in ascending powers of x, up to and including the term in x5 Then use your expansion to find an approximation for sin10˚

FM Maclaurin Taylor series name _______________________

WB7 write down the first 4 non-zero terms in the series expansion, in ascending powers of x, of f ( x )=cos (2 x2 )

WB8 write down the first 4 non-zero terms in the series expansion, in ascending powers of

x, off ( x )=¿ ln {√1+2 x1−3 x }

FM Maclaurin Taylor series name _______________________

WB9 Given that terms in xn, n > 4 can be ignored, show, using the series expansions of ex

and sinx, that: esinx≈1+ x+ x2

2− x

4

8

FM Maclaurin Taylor series name _______________________

Taylor Series

f ( x+a )=f (a )+f ' (a ) x+ f ' '(a)2 !

x2+ f ' ' '(a)3!

x3…

f ( x )=f (a )+ f ' (a )(x−a)+ f ' ' (a)2!

( x−a)2+ f ' ' ' (a)3 !

(x−a)3…

Both these forms can be used under different circumstances as Taylor expansionsYou will need to decide which to use based on the question!

WB10 Find the Taylor expansion of ln x in powers of (x -1) up to and including the terms in (x−1)3

FM Maclaurin Taylor series name _______________________

WB11 Find the Taylor expansion of ln x in powers of (x - 1) up to and including the terms in (x - 1)3

WB12 Find the Taylor expansion of e-x in powers of (x + 4) up to and including the terms in (x + 4)3

FM Maclaurin Taylor series name _______________________

WB13 Express: tan(x+ π4 ) As a series in ascending powers of x, up to the term in x3

WB14 a) Show that the Taylor expansion of sin x in ascending powers of (x – π/6) up to the

term in (x – π/6)2 is: sinx= 12+ √32 (x−π6 )−14 ( x−π6 )

2

b) Then use the series to find an approximation for sin40, in terms of π

FM Maclaurin Taylor series name _______________________

FM Maclaurin Taylor series name _______________________

WB15 Show that the Taylor expansion of sin (x+ π6 )up to the term in x3 is:

12+ √32x−14x2−√3

12x3+….

WB16 Given that y=x ex

a) Show that dn yd xn

=(n+ x )ex

b) Find the Taylor expansion of y=x ex in ascending powers of ( x+1 ) as far as the term in ( x+1 )4

FM Maclaurin Taylor series name _______________________

We can adapt the Maclaurin series to find solutions of differential equations

By substituting given values into a differential equation, and differentiating the DE to find higher order derivatives, the coefficients can be found and ‘plugged into’ the standard form

WB17 use the Taylor method to find a series solution, in ascending powers of x, up to and

including the term in x3, of d2 yd x2

= y−sin x (1)

given that when x=0 , y=1 , and dydx

=2

FM Maclaurin Taylor series name _______________________

WB18 (1+2 x ) dydx

=x+4 y2 (1)

a) Show that (1+2 x ) d2 yd x2

=1+2 (4 y−1 ) dydx

(2)

b) Differentiate equation (1) with respect to x to obtain an equation

involving d2 yd x2

d2 yd x2

, d2 yd x2

, x and y

given that when x=0 , y=12c) Find a series solution for y, in ascending powers of x, up to and including the term in x3

FM Maclaurin Taylor series name _______________________