Sage Rat

01-16-2006, 12:59 AM

I'm teaching myself the calculus from Calculus Made Easy (1998 ed.) I've only just begun, and have been thrown off by a bit of simple math on page 63. Thompson shows us two examples of determining for what length of h and r where h=r, a one inch difference in size will cause a 400 in.3 difference in volume of a cylinder.

The first example is where h is equal to r initially but remains a constant (i.e. the one inch variation will only be used on r.) To show this he pulls the equation dV/dr = 2πr2 out of his butt. Then for the equation where h and r will both vary by 1 inch to create the 400 in.3 difference he uses dV/dr = 3πr2. Neither of these does he show where he is coming to these equations.

V = πr2h is the original equation which he would have started from to create the derivation. If h = r and varies with it, I would think this would be V = πr2*r, which becomes πr3. Then derived it would be dV/dr = 2πr2--but that's the case where h is supposed to remain as a constant, while as I had expected it to be the case where h varies.

The first example is where h is equal to r initially but remains a constant (i.e. the one inch variation will only be used on r.) To show this he pulls the equation dV/dr = 2πr2 out of his butt. Then for the equation where h and r will both vary by 1 inch to create the 400 in.3 difference he uses dV/dr = 3πr2. Neither of these does he show where he is coming to these equations.

V = πr2h is the original equation which he would have started from to create the derivation. If h = r and varies with it, I would think this would be V = πr2*r, which becomes πr3. Then derived it would be dV/dr = 2πr2--but that's the case where h is supposed to remain as a constant, while as I had expected it to be the case where h varies.