It is a beautiful Spring Afternoon, you are sitting in the park, and you are happy. Your favorite Volleyball team is going to the Olympics, but that isn't what is making you happy. Your sweetheart has just told you that you are getting a nice surprise for May Day, but that isn't what is making you happy. A magazine company has just informed you that you are entered in a million dollar sweepstakes, but that isn't what is making you happy. You are happy because you have learned a new theorem:

If dy/dx = k y, then y = Aekx.

The nice thing about the theorem, you think, is that the converse is also true. In other words:

If y = Aekx then dy/dx = k y.

You hear some tinkly bells, and there is the ice-cream cart! You wave, and it comes towards you. But there is something wrong with the ice-cream vendor. He is wearing large dark glasses and an obviously fake beard and mustache. You buy the ice-cream anyway (one of those ice cream bars on a stick), but after he has gone away, you notice this written on the wrapper.

Proof of the Theorem you learned today

dy/dx = k y

dy / y = k dx

Integration gives ln y = kx + c

y = e kx + c

y = ecekx

y = Aekx

There are two unusual things going on here. First of all, this message appeared on your wrapper in the first place. Second of all, you are most of the way through a "Find the error" problem, and there isn't an error here. In fact, you might have seen this very proof in your calculus class. You eat your ice-cream, and then notice this written on the stick:;

Since A = ec, then A can't be zero, and A can't be negative. But if y = Aekx then dy/dx = k y, no matter what the value of A. So why didn't your teacher's 'proof' catch all the solutions?

Why indeed? Find the error.

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