Integration by parts and approximation by power series

In summary, when finding v(t) and x(t) for an object subject to a time-dependent force, it is important to develop the exponential term up to t^2 before integrating, but after integrating, it is sufficient to include only up to the t term. This is because integration increases the order of t by one.
  • #1
Elvis 123456789
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6

Homework Statement



An object of mass m is initially at rest and is subject to a time-dependent force given by F = kte^(-λt), where k and λ are constants.
a) Find v(t) and x(t).
b) Show for small t that v = 1/2 *k/m t^2 and x = 1/6 *k/m t^3.
c) Find the object’s terminal velocity.

Homework Equations

The Attempt at a Solution


I showed my work for finding v(t) and getting the approximation for v(t) for small t. but I am missing the 1/2 at the front and I can't seem to find where it comes from.
 

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  • #2
Perhaps you need to further develop ##e^{-\lambda t}## ? The term linear in ##t## cancels, so you need to develop up to ##t^2##
 
Last edited:
  • #3
BvU said:
Perhaps you need to further develop ##e^{-\lambda t}## ? The term ilinear in ##t## cancels, so you need to develop up to ##t^2##
but if I include a t^2 term then I end up with a t^3 term that doesn't cancel
 
  • #4
Yes. So the outcome will be correct up to order ##t^2##. .
 
  • #5
You can also do the development before the integration: ## t\;e^{-\lambda t}= t -\lambda t^2 ## and ignore higher orders
 
  • #6
BvU said:
You can also do the development before the integration: ## t\;e^{-\lambda t}= t -\lambda t^2 ## and ignore higher orders
Thanks for your help, i got the right answer. But can you help me understand something, why is it enough to include only up to the t^1 term before the integration, but after the integration we have to include the t^2 term?
 
  • #7
"Integration increases order of ##t## by one" is the answer that comes to mind. But I agree with you that it makes a weird impression.
 

1. What is integration by parts and how is it used in calculus?

Integration by parts is a technique used in calculus to find the integral of a product of two functions. It involves using the product rule of differentiation in reverse, where one function is differentiated and the other is integrated. This method is particularly useful when the integral involves a product of functions that cannot be easily integrated by other methods.

2. Can integration by parts be used to solve definite integrals?

Yes, integration by parts can be used to solve definite integrals. After applying the integration by parts formula, the resulting integral can be evaluated using the limits of integration to find the definite integral. This can be helpful in solving integrals that are not easily evaluated using other techniques.

3. How does approximation by power series work?

Approximation by power series is a method used to approximate a function using a polynomial with an infinite number of terms. This is done by representing the function as a sum of powers of x and then finding the coefficients of each term using the Taylor series expansion. The more terms that are included in the polynomial, the more accurate the approximation will be.

4. What is the difference between Taylor series and Maclaurin series?

Taylor series and Maclaurin series are both methods of approximating a function using a polynomial with infinitely many terms. The main difference is that Maclaurin series is a special case of the Taylor series, where the center of the approximation is at x=0. In other words, Maclaurin series is a Taylor series with a=0.

5. When should I use integration by parts and when should I use approximation by power series?

Integration by parts should be used when trying to find the integral of a product of two functions. It is also helpful when trying to solve definite integrals that cannot be evaluated using other techniques. Approximation by power series should be used when trying to approximate a function using a polynomial with infinitely many terms. This method is particularly useful when trying to find an approximation of a function that cannot be easily evaluated or simplified.

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