Taylor Approximation: Show ∫f'(x)dx/f(x)=ln|f(x)|+C

In summary: There is nothing wrong with using the Fundamental Theorem of Calculus to solve the problem. To verify that ##\int f(x) \, dx = F(x) + C##, just check that ##F'(x) = f(x)##. That is an absolutely 100% correct way to do the question. In fact, such checking should always be done out of habit, whenever you are faced with a possible formula for the indefinite integral F(x) of an integrand f(x). That is a good way to catch errors.In summary, Ray suggests that Taylor approximation may be a good way to approach the homework question. He provides derivatives and verification that the answer is indeed correct.
  • #1
Nugso
Gold Member
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Homework Statement



Show that [tex]∫f'(x)dx/f(x) = ln|(f(x)|+C[/tex] where f(x) is a differential function.


Homework Equations



First order Taylor approximation? [tex]f(x)=f(a)+f'(a)(x-a)[/tex]


The Attempt at a Solution



Well, I'm not really sure how to approach the question. It's my Numerical Methods homework, so I think I have to do it by using Taylor approximation. By applying the first order Taylor approximation I get:

[tex]ln|f(x)|=y, ln|f(a)| - ln|(f(x)| = (a-x)f'(x)/f(x)[/tex]

[tex]ln|(f(a)/f(x)| = (a-x)f'(x)/f(x)[/tex]

I'm kind of stuck here. Am I right in thinking that Taylor approximation is an appropriate way to approach the question?
 
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  • #2
Nugso said:

Homework Statement



Show that [tex]∫f'(x)dx/f(x) = ln|(f(x)|+C[/tex] where f(x) is a differential function.


Homework Equations



First order Taylor approximation? [tex]f(x)=f(a)+f'(a)(x-a)[/tex]


The Attempt at a Solution



Well, I'm not really sure how to approach the question. It's my Numerical Methods homework, so I think I have to do it by using Taylor approximation. By applying the first order Taylor approximation I get:

[tex]ln|f(x)|=y, ln|f(a)| - ln|(f(x)| = (a-x)f'(x)/f(x)[/tex]

[tex]ln|(f(a)/f(x)| = (a-x)f'(x)/f(x)[/tex]

I'm kind of stuck here. Am I right in thinking that Taylor approximation is an appropriate way to approach the question?

Start again: you are on the wrong track.
 
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  • #3
Hi Ray, thanks for the reply. Would you mind giving me a hint? I really haven't a clue about the right track.
 
  • #4
Why not take the derivative of the right-hand side?
 
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  • #5
micromass said:
Why not take the derivative of the right-hand side?

I guess my teacher wants us to solve it by the difficult way, i.e using appropriate ways.
 
  • #6
Nugso said:
I guess my teacher wants us to solve it by the difficult way, i.e using appropriate ways.

There is nothing wrong with using the Fundamental Theorem of Calculus to solve the problem. To verify that ##\int f(x) \, dx = F(x) + C##, just check that ##F'(x) = f(x)##.

That is an absolutely 100% correct way to do the question. In fact, such checking should always be done out of habit, whenever you are faced with a possible formula for the indefinite integral F(x) of an integrand f(x). That is a good way to catch errors.
 
  • Like
Likes 1 person

1. What is Taylor Approximation?

Taylor Approximation is a mathematical technique used to approximate a complicated function using a simpler one. It is based on the Taylor series, which breaks down a function into an infinite sum of polynomial terms.

2. What is the significance of the equation ∫f'(x)dx/f(x)=ln|f(x)|+C?

This equation is known as the logarithmic form of Taylor's theorem. It is used to approximate the value of a function at a specific point by integrating its derivative and taking the natural logarithm of the function. The constant C represents the error term in the approximation.

3. How is Taylor Approximation used in real-world applications?

Taylor Approximation is used in many fields of science, including physics, engineering, and economics. It is used to approximate the behavior of physical systems, make predictions, and solve differential equations. It is also used in data analysis and modeling to simplify complex data sets and make predictions.

4. What are the limitations of Taylor Approximation?

The main limitation of Taylor Approximation is that it can only provide accurate approximations for functions that are smooth and continuous. It also becomes less accurate as the number of terms in the Taylor series increases. Additionally, it can only approximate functions within a certain radius of convergence.

5. Are there any alternatives to Taylor Approximation?

Yes, there are other methods of function approximation, such as linear approximation, spline interpolation, and Fourier series. These methods may be more suitable for certain types of functions or situations, and it is important to choose the appropriate method based on the specific problem at hand.

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