How can I find x in terms of c for this expression?

  • Thread starter grepecs
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In summary: I forgot to add the constant to the right side of the equation. In summary, x=ab tanh(b/c) is the correct expression for x in terms of c.
  • #1
grepecs
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Homework Statement


With

[tex]f(x)=-\lgroup\frac{a}{2}-\frac{x}{2b}\rgroup\ln\lgroup\frac{a}{2}-\frac{x}{2b}\rgroup-\lgroup\frac{a}{2}+\frac{x}{2b}\rgroup\ln\lgroup \frac{a}{2}+\frac{x}{2b}\rgroup[/tex]

and

[tex]z=\frac{df}{dx}=\frac{1}{c}[/tex]

find an expression of x in terms of c.

Homework Equations


Well, relevant should be that the answer is supposed to be

[tex]x=ab \tanh\lgroup\frac{b}{c}\rgroup[/tex]

The Attempt at a Solution



Differentiating f wrt x, I get

[tex]\frac{df}{dx}=-\frac{1}{2b}\lgroup \ln\lgroup\frac{a}{2}-\frac{x}{2b}\rgroup+\ln \lgroup\frac{a}{2}+\frac{x}{2b}\rgroup+2\rgroup[/tex]

But this is no good, because when I exponentiate on both sides of the equation (in order to solve for x)

[tex]-\frac{2b}{c}-2=-2b\frac{df}{dx}-2[/tex]

I end up with a something like

[tex]x=2b\sqrt\lgroup -\exp(-2\frac{b}{c}-2)-(\frac{a}{2})^2\rgroup,[/tex]

which is obviously incorrect.

I'd be happy if someone would help me.
 
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  • #2
grepecs said:

Homework Statement


With

[tex]f(x)=-\lgroup\frac{a}{2}-\frac{x}{2b}\rgroup\ln\lgroup\frac{a}{2}-\frac{x}{2b}\rgroup-\lgroup\frac{a}{2}+\frac{x}{2b}\rgroup\ln\lgroup \frac{a}{2}+\frac{x}{2b}\rgroup[/tex]

and

[tex]z=\frac{df}{dx}=\frac{1}{c}[/tex]

find an expression of x in terms of c.

Homework Equations


Well, relevant should be that the answer is supposed to be

[tex]x=ab \tanh\lgroup\frac{b}{c}\rgroup[/tex]

The Attempt at a Solution



Differentiating f wrt x, I get

[tex]\frac{df}{dx}=-\frac{1}{2b}\lgroup \ln\lgroup\frac{a}{2}-\frac{x}{2b}\rgroup+\ln \lgroup\frac{a}{2}+\frac{x}{2b}\rgroup+2\rgroup[/tex]

But this is no good, because when I exponentiate on both sides of the equation (in order to solve for x)

[tex]-\frac{2b}{c}-2=-2b\frac{df}{dx}-2[/tex]

I end up with a something like

[tex]x=2b\sqrt\lgroup -\exp(-2\frac{b}{c}-2)-(\frac{a}{2})^2\rgroup,[/tex]

which is obviously incorrect.

I'd be happy if someone would help me.

Did you use the product rule when you differentiated? I didn't check your work very closely, but it doesn't seem that you did.
 
  • #3
Mark44 said:
Did you use the product rule when you differentiated? I didn't check your work very closely, but it doesn't seem that you did.

I think I did. The derivative of

[tex]\lgroup\frac{a}{2}+\frac{x}{2b}\rgroup\ln\lgroup \frac{a}{2}+\frac{x}{2b}\rgroup[/tex]

is, according to my calculations,

[tex]\frac{1}{2b}\ln\lgroup \frac{a}{2}+\frac{x}{2b}\rgroup+\lgroup\frac{a}{2}+\frac{x}{2b}\rgroup\frac{1}{\frac{a}{2}+\frac{x}{2b}}\frac{1}{2b}[/tex]

The second term reduces to 1/2b, which is broken out of the expression.
 
  • #4
No one who has any suggestions? I'm pretty sure it's a pretty simple error :)
 
  • #5
Ok, mr. Wolfram Alpha solved the problem for me (quite the dude, isn't he?).
 

1. What is an expression in terms of science?

An expression in science is a mathematical statement that represents a relationship between variables. It can include numbers, symbols, and mathematical operations to describe a specific phenomenon or process.

2. Why is finding an expression important in scientific research?

Finding an expression is important in scientific research because it allows for the quantification and understanding of complex systems and phenomena. It also enables scientists to make predictions and test hypotheses.

3. How do scientists go about finding an expression?

Scientists use a combination of observation, experimentation, and mathematical analysis to find an expression. They first observe a phenomenon or process, then conduct experiments to collect data, and finally use mathematical tools to analyze the data and create an expression.

4. Can an expression be changed or modified?

Yes, an expression can be changed or modified depending on new data or information. As scientists learn more about a particular phenomenon, they may refine or revise the expression to better represent the relationship between variables.

5. Are there different types of expressions in science?

Yes, there are different types of expressions in science, such as algebraic expressions, chemical expressions, and statistical expressions. Each type serves a specific purpose and is used to describe different types of relationships between variables.

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