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What is the antiderivative of the distance function?

Answer 1

Luke Alvarez

The distance function is:

#D = sqrt((Deltax)^2 + (Deltay)^2)#

Let's manipulate this.

#= sqrt((Deltax)^2 + (Deltay)^2/(Deltax)^2 (Deltax)^2)#

#= sqrt(1 + (Deltay)^2/(Deltax)^2)Deltax#

Since the antiderivative is basically an indefinite integral, this becomes an infinite sum of infinitely small #dx#:

#= sumsqrt(1 + (Deltay)^2/(Deltax)^2)Deltax#

#= int sqrt(1 + ((dy)/(dx))^2)dx#

which happens to be the formula for the arc length of any function you can manageably integrate after the manipulation.

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Answer 2

Ezra Adams

The antiderivative of the distance function would be the displacement function.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.