First Order Error Analysis (Taylor Series)

[jegues]

Homework Statement

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

$$v(t) = \frac{gm}{c}(1-e^{-(\frac{c}{m})t})$$

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)

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

Thanks again!

 

[LCKurtz]

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.

 

[jegues]

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.