Taylor series application

In summary, the conversation discusses the velocity of a water wave and how it is affected by the depth of the water. It presents two cases, one where the water is deep and one where it is shallow, and provides equations for calculating the velocity in each case using the MacLaurin series for tanh. The conversation also mentions the use of the variables L and d to determine whether the water is deep or shallow.
  • #1
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Homework Statement


A water wave has length L moves with velocity V across body of water with depth d, then v^2=gL/2pi•tanh(2pi•d/L)
A) if water is deep, show that v^2~(gL/2pi)^1/2
B) if shallow use maclairin series for tanh to show v~(gd)^1/2

Homework Equations



Up above

3. The Attempt at a Solution

Have no idea where to start, just looking for some tips and I'm assuming I need the tanh series for both parts, this has been the hardest calculus for me
 
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  • #2
Well, first things first: define "deep" and "shallow". In each case it means that one of your variables is much less, or much greater, than another. Figure out which quantity is less/more than which other one in each case, and that should tell you what variable or combination of variables to use for your series expansion.

Also, you can look up the series expansion for tanh(x) on Wikipedia, among other places.
 

What is a Taylor series?

A Taylor series is an infinite sum of terms that represents a function. It is used to approximate a function by adding up terms that represent the value of the function and its derivatives at a specific point.

How is a Taylor series useful in real-life applications?

Taylor series are used in many different fields of science and engineering, such as physics, chemistry, and economics. They are useful for approximating complicated functions and solving differential equations.

What is the formula for a Taylor series?

The formula for a Taylor series is f(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2! + f'''(a)(x-a)^3/3! + ... where f(x) is the function, a is the point of approximation, and f'(a), f''(a), f'''(a), etc. are the derivatives of the function evaluated at a.

What is the difference between a Taylor series and a Maclaurin series?

A Taylor series is a generalization of the Maclaurin series, which is a Taylor series centered at a = 0. In other words, a Maclaurin series is a special case of a Taylor series where the point of approximation is 0.

How do you determine the accuracy of a Taylor series approximation?

The accuracy of a Taylor series approximation depends on the number of terms used in the series. The more terms used, the more accurate the approximation will be. Additionally, the error of a Taylor series approximation can be estimated using the remainder term formula, which takes into account the value of the next term in the series.

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