First Order Error Analysis (Taylor Series)

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Homework Statement



The equation for the velocity of a falling parachutist can be computed by,

[tex]v(t) = \frac{gm}{c}(1-e^{-(\frac{c}{m})t})[/tex]

Use a first-order error analysis to estimate the error of v at t = 6, if g = 9.8 and m = 9.8, c = 12.5 plus or minus 1.5.

Homework Equations





The Attempt at a Solution



I've never done "first order error analysis" using taylor series so I've looked at the solution to try and do the problem backwards but I can't make sense of what's going on. (See figure attached)

attachment.php?attachmentid=31519&stc=1&d=1295722310.jpg


Can someone clarify to me what they're doing here? I need the ideas behind what's being done.

Thanks again!
 

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  • #2
LCKurtz
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He is using approximation by differentials. In one variable, say you have a function f(x) and you know it and its derivative at x0. You can approximate its value at some nearby point x1 by

f(x1) ≈ f(x0) + f'(x0)(x1-x0)

You may see it written as

f(x1) - f(x0) ≈ f'(x0)(x1-x0)

or

Δy ≈ f'(x)Δx

For example, the second line in your image is exactly this last equation with y and x replaced by v and c.
 
Last edited:
  • #3
1,097
2
He is using approximation by differentials. In one variable, say you have a function f(x) and you know it and its derivative at x0. You can approximate its value at some nearby point x1 by

f(x1) ≈ f(x0) + f'(x0)(x1-x0)

You may see it written as

f(x1) - f(x0) ≈ f'(x0)(x1-x0)

or

Δy = f'(x)Δx

For example, the second line in your image is exactly this last equation with y and x replaced by v and c.
EDIT: I think I got it now.
 

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