Calculus/Continuity

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Expert: - 2/12/2009


Question

question #1 picture
A swimmer crosses a river starting at point A and ending at point B, following the path shown below. Prove
that for some value x, the swimmer's distance d (Px, A) from A is the same as the distance d (Px, B) from
B.

Answer
Questioner:  Mona
Private: no

A swimmer crosses a river starting at point A and ending at point B, following the path shown below. Prove that for some value x, the swimmer's distance d (Px, A) from A is the same as the distance d (Px, B) from B.  

............................................
This is a matter of giving names to your variables -- the things that change. (you will want to do this when you get to Related Rates and Max-min problems, so you might as well start now.)


Let s = the arc length along the path from A to B. (actual swimming distance)
   x = the horizontal distance from a, and a function of s
   PA = the actual distance of the swimmer from A. also a function of s.
   PB = the actual distance of the swimmer from B. also a function of s.
   D = AB, the distance from A to B (a constant)


Observe that as the swimmer swims, s always increases.
Also, x always increases.
Also, the swimmer never 'jumps', so x(s) is continuous.
So is PA(s), and so is PB(s)

PA(s) starts at  0, and finishes with a value of  D.
PB(s) starts at D, and finishes with a value pf 0.

Let Diff(s) = PA(s) - PB(s).

Diff(s) starts at 0 - D = -D.
Diff(s) finishes at D - 0 = +D.

So Diff(s) is a continuous function of s with values that include +D and -D.

You take it from there.

Paul Klarreich

Expertise

All topics in first-year calculus including infinite series, max-min and related rate problems. Also trigonometry and complex numbers, theory of equations, exponential and logarithmic functions. I can also try (but not guarantee) to answer questions on Analysis -- sequences, limits, continuity.

Experience

I taught all mathematics subjects from elementary algebra to differential equations at a two-year college in New York City for 25 years.

Education/Credentials
(See above.)

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