Impossible math question?

When I was in class today, I found out I couldnt answer a question without my graphical calculator.
But, I want to know, is there anyone with a method to solve this problem without one?

x * (e ^(0.5x)) = 0.5

Is there a way to find the value of x?

Using intersect with graphs, its possible to find the value, but I want to do it without intersect :frown:

Please help me,
Seph

Scroll to see replies

Reply 1

north bank gooner

You could try converting both sides into logs. Sorry I can't help more, it's been a while since I last touched a maths textbook.

Reply 2

Mokert23

The answer is 1. Sorry no help

Reply 3

Dystopia

Sephirona

Please help me,
Seph

There is a way, but not using techniques learnt at A-Level; you would have to use the Lambert W function, which is the inverse function of

f(x) = xe^{x}

, in the same way that

\ln x

is the inverse function of

e^{x}

. If

y = xe^{x}

, then

x = W(y)

, which is defined by an infinite series:

\displaystyle W(y) = \sum_{n=1}^{\infty} \frac{(-n)^{n-1}}{n!} y^{n}

So, in your example,

0.5xe^{0.5x} = 0.25

, so

x = 2W(0.25)

You can evaluate that here.

Reply 4

Dadeyemi

not to hard to use Newton-Raphson on this i think. doesn't fixed point iteration work if you take logs?

Reply 5

Creole

So basically, non-solvable using elementary functions, so you have to use numerical methods at A-Level.

Reply 6

jc123

i think i done it but im guessing its probably wrong as ur all saying its impossible, the working is attached

maths.jpg11.3KB

Reply 7

generalebriety

jc123

i think i done it but im guessing its probably wrong as ur all saying its impossible, the working is attached

First mistake (of rather a few):

e^{0.5x} = 0.5/x \implies \ln (0.5/x) = 0.5x

, not what you wrote.

Reply 8

Sephirona

So, The only way to solve is is to do a series of commands that gets closer to the answer?

So, no exact values?.
I guess there's no point in doing this then. I'm guessing my graphical calculator does the exact same thing.

Nonetheless, thanks a lot you both for explaining me these methods. I'll most certainly will be able to use them tomorrow, in my class where I'm not allowed to use any calculators.

I guess 2W(0.25) is an exact answer, or at least, as exact as it gets (Because of the infinity)

Reply 9

Gaz031

I guess 2W(0.25) is an exact answer, or at least, as exact as it gets (Because of the infinity)

Expressing it as an infinite series doesn't stop it being an exact answer. For example, the exponential function e^x can be written as an infinite series which defines it exactly. :smile:

/pedant.

Reply 10

Dadeyemi

x=0.4077767094 with fixed point iteration not exactly a beautiful solution but should be correct. by examining the graph of y= 0.5/x and y=e^(0.5x) I believe we can show this is the only solution.

edit: for a fuller proof y= 0.5/x is negative when x<0 x=/=0 and is strictly decreasing when x > 0. and y=e^(0.5x) is always positive for all real x and strictly increasing.

Reply 11

Dystopia

Gaz031

Expressing it as an infinite series doesn't stop it being an exact answer. For example, the exponential function e^x can be written as an infinite series which defines it exactly. :smile:

/pedant.

Indeed; while the infinite series looks rather daunting, it is still exact. Calculators will use infinite series to evaluate trigonometric functions, logarithms and so on. The only difference is that it is not an elementary function (basically, you won't find it on your calculator...).

Reply 12

maths-enthusiast

@threadstarter: except using numerical methods for approximate answers, you can't do much.

@dystopia, mate can you evaluate let's say e^2 exactly? just because there is an infinite series representation and it converges doesn't mean you can calculate it in finite time. and i think, that's what threadstarter meant when he/she said this problem doesn't give you an exact answer. irrational answers wasn't what she expected.

Reply 13

Dystopia

maths-enthusiast

@dystopia, mate can you evaluate let's say e^2 exactly? just because there is an infinite series representation and it converges doesn't mean you can calculate it in finite time. and i think, that's what threadstarter meant when he/she said this problem doesn't give you an exact answer. irrational answers wasn't what she expected.

No; as you say, it is irrational. Although, using the series expansion I can get an answer to any desired accuracy. However, I wasn't talking about the decimal approximation when I referred to an 'exact answer'. I would argue that the answer

x = 2W(0.25)

is just as exact as the answer

x = e^{2}

is if one were trying to solve

\ln x = 2

.

I'm not sure what you're trying to argue. Those solutions are, by definition, exact.

Reply 14

maths-enthusiast

mate, irrational numbers are not 'exact' in the sense that they don't terminate. that's what i meant when i said the threadstarter wasn't looking for an irrational answer, he/she thought the question had an 'exact' or rational answer.

Reply 15

Jake22

maths-enthusiast

What about a rational number like

\frac{1}{3}

. The decimal representation of that never terminates... would you consider that to also be 'un-exact' ?? You can't represent that as a decimal in finite time.

I think your use of the term 'exact' is highly non-standard.

Reply 16

qgujxj39

"e" is a constant, i.e. it is exact.
"e" is also irrational.

Hmm.

Reply 17

Chewwy

maths-enthusiast

this is stupid. by your argument:

x^2+1=0 has no exact answer
x^2-2=0 has no exact answer
2sinx-1=0 has no exact answer
...

Reply 18

Chewwy

tommmmmmmmmm

"e" is a constant, i.e. it is exact.
"e" is also irrational.

Hmm.

not seeing the logic here.

0.5 is an 'unexact' (i.e. approximate) solution to the OP's equation, and yet is a certainly constant, hence there certainly isn't equivalence between something being constant, and something being an exact solution.

this whole discussion is ridiculous anyway. does 'exact' mean constant? no. does 'exact' mean rational? no. all it means is that we can write our solution in some closed form.

Reply 19

Dystopia

I somehow get the impression that even if every single member of TSR explained in detail why maths-enthusiast was wrong, he would continue to maintain that he was correct (while calling us all 'mate').

Sorry, but you are rather annoying...