Estimate the final position using the given velocity function and initial position.

Estimate the final position using the given velocity function and initial position.

v(t) = 30e^{-t/4 }, s(0) = -1, b = 4

I have been looking in my book and it is of no help really. Only two examples in their and not even really close to explaining what I am supposed to do. Any help is appreciated.

P.S. Hopefully ya'll do not get tired of my constant barrage of questions. (Itwasntme)

Re: Estimate the final position using the given velocity function and initial positio

What you are probably expected to do is use the differential to get a linear approximate.

$\displaystyle \frac{\Delta s}{\Delta t}\approx\frac{ds}{dt}$

$\displaystyle \Delta s\approx\frac{ds}{dt}\cdot\Delta t$

$\displaystyle s(t)-s(0)\approx\frac{ds}{dt}\cdot\Delta t$

Now, using $\displaystyle v\equiv\frac{ds}{dt}$ we have:

$\displaystyle s(t)\approx v(0)\cdot\Delta t+s(0)$

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Re: Estimate the final position using the given velocity function and initial positio

Thanks for the quick reply...However, I am really drawing a blank here... in the book, we are discussing the Definite Integral, and the closest example I can find to this problem is one for "A Midpoint Rule Approximation of a Definite Integral".

I am attaching a copy of that example from the book, in a picture file. Perhaps I am simply over thinking this? (Worried)

Re: Estimate the final position using the given velocity function and initial positio

okay, so as an update, I looked at an odd problem in the book, that was almost the same as this one..... and it looks like I need to compute the (definite integral) 30e^{-t/4} dt

I am assuming I do this by computing various right hand sums for different values of n.

however the 30e^{-t/4} is kind of throwing me off and not sure how to approach this. (Headbang)

Re: Estimate the final position using the given velocity function and initial positio

There are many numeric schemes for approximating the solutions to ODEs, which have their analogs to approximating definite integrals via the FTOC. For your problem, you may state:

$\displaystyle s(4)=30\int_0^4e^{-\frac{t}{4}}\,dt-1$

Now, you may use Riemann sums, the Trapezoidal Rule, Simpson's Rule, or Newton-Cotes methods, just to name a few. To just tell you to approximate the final position leaves a lot of leeway on your part to choose the method and number of partitions.

Re: Estimate the final position using the given velocity function and initial positio

I know I need (should) use Riemann sums, but I am having difficulty in my confidence to set it up correctly.

This is how I set it up:

$\displaystyle An = 30({\frac{4}{n}}\sum_0^ne^{-\frac{4i}{4n}}\ dt - 1)$

Does this look to be setup correctly?

Re: Estimate the final position using the given velocity function and initial positio

Using a left sum to approximate just the integral, I would set it up as:

$\displaystyle \Delta t=\frac{4-0}{n}=\frac{4}{n}$

$\displaystyle t_k=0+k\Delta t=k\frac{4}{n}$

$\displaystyle A_n=\frac{4}{n}\sum_{k=0}^{n-1}e^{-\frac{k}{n}}$