quadratic approximations (partial fractions)

hey, i am trying to do this but im not sure what i am doing wrong, although i have some idea where i might have went wrong whilst i typing this, but i'd rather get the help whilst having ago at this again myself.

i) i had to express 7-4x / (2x-1)(x+2) in partial fractions as A/(2x-1) +B/(x+2). I worked out that A=2 and B=-3.

ii) i had to express 1/(1-2x) in the form of a+bx+cx^2, which i did binomial. i got 1+2x+4x^2, range |x|<1/2

iii) i had to do the same for 1/(2+x). i got 1/2 - x/4 + x^2/8

now im trying to do this. i'll quote the question

"Hence find a quadratic approximation for 7-4x/(2x-1)(x+2) when |x| is small. Find the relative error in this approximation when x=0.1"

what i did was multiply ii) by 2 and iii) by -3, but i didn't get the answer, however i think the 2x in (2x-1) has some effect on this, although im not sure. The answer is meant to be -7/2 - 13x/4 - 67x^2/8 , error of 0.505%. anyways im gonna have another go whilst watching the football. if i figure it out i'll post the answer. thanks in advance if anyone does help me :smile:

Reply 1

Coolraj

5

jones001

hey, i am trying to do this but im not sure what i am doing wrong, although i have some idea where i might have went wrong whilst i typing this, but i'd rather get the help whilst having ago at this again myself.

i) i had to express 7-4x / (2x-1)(x+2) in partial fractions as A/(2x+1) +B/(x+2). I worked out that A=2 and B=-3.

ii) i had to express 1/(1-2x) in the form of a+bx+cx^2, which i did binomial. i got 1+2x+4x^2, range |x|<1/2

iii) i had to do the same for 1/(2+x). i got 1/2 - x/4 + x^2/8

for part i) it's A/2x-1 + B/x+2

for part ii) 1/2+x
= (2 + x)^-1
= 2^-1(1+x)^-1
do bionomial expansion on that...
hope that helped :biggrin:

EDIT: just saw the other part :colondollar:

just add the results you got in ii) and iii). to find the error, i think you firstly you need to substitute x = 0.1 into the original equation and then work out an approximation using the bionomial expansion (which you've done previously) and divide approximation by original and multiply by 100...

Reply 2

steve2005

17

jones001

hey, i am trying to do this but im not sure what i am doing wrong, although i have some idea where i might have went wrong whilst i typing this, but i'd rather get the help whilst having ago at this again myself.

i) i had to express 7-4x / (2x-1)(x+2) in partial fractions as A/(2x+1) +B/(x+2). I worked out that A=2 and B=-3.

ii) i had to express 1/(1-2x) in the form of a+bx+cx^2, which i did binomial. i got 1+2x+4x^2, range |x|<1/2

iii) i had to do the same for 1/(2+x). i got 1/2 - x/4 + x^2/8

now im trying to do this. i'll quote the question

"Hence find a quadratic approximation for 7-4x/(2x-1)(x+2) when |x| is small. Find the relative error in this approximation when x=0.1"

what i did was multiply ii) by 2 and iii) by -3, but i didn't get the answer, however i think the 2x in (2x-1) has some effect on this, although im not sure. The answer is meant to be -7/2 - 13x/4 - 67x^2/8 , error of 0.505%. anyways im gonna have another go whilst watching the football. if i figure it out i'll post the answer. thanks in advance if anyone does help me :smile:

Here is the first part.

Picture 133.png35.5KB

Reply 3

jones001

OP

yeh did it at the end, did it after i posted, had a look in the text book. cheers anyways.

quadratic approximations (partial fractions)

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