x=f(p)=(264-2p^2)^3/2
I need to find the elasticity of demand at a given point, p.
E(p)= -pf`(p)/f(p), right?
When I derive f(p), I get 3/2(264-2p^2)^1/2*-4p, which simplifies to -6p(264-2p^2)^1/2. Is this as simplified as it can go? I am so lost as to where to start with this...
Is this a business calculus class? I have never heard of elasticity of demand. Sounds very interesting though.
Your derivative looks correct. You should just plug your f'(p) back into the elasticity equation and it should simplify really easily. Something like this:
-(p)*(-6p(264-2p^2)^1/2)*((264-2p^2)^-3/2)
Because the functions inside the exponential terms for the numerator and denominator are the same you can treat them as common variables and look at it like this
-(p)*(-6P(X)^1/2)*((X)^-3/2)
Should be very easy to simplify from there and you just plug in X = (264-2p^2) and you dont even need to distribute that 6p^2 in the front.
great, now I feel really stupid...
figured it out. f'(p)p/f(p) simplifies to 3p^2/134-p^2.
Yeah it's business calc. Lots of word problems. Pain in the butt. Thanks for the help :)
the elasticity of demand is just deriving the price/demand function and estimating the rate of change of demand when price is changed by a given percentage.
If p(x) is the price/demand function, then x=f(p).
The furthest I am able to simplify that is this
E(p) = (-6p^2)*((264-2p^2)^-1/2)
I am not sure how you extracted the two from the bottom or got it out of the square root but I could be missing something. That is really cool, instantaneous demand. I never knew how much they actually applied calculus to economics. I should take these classes over again. :D
EDIT: Ahh no your right. I don't know what I am thinking, that exponent does reduce to one. Apparently I cant subtract 1/2 from 3/2 anymore ;D. Yeah it looks like your final result is good!
when you derive (264-2p^2)^3/2, you get -6p(264-2p^2)^1/2. You then take that and put it into the elasticity formula, which is p*f(p)/f(p), and divide by (264-2p^2)^3/2. You subtract 1/2 by 3/2 (dividing exponents) to get (264-2p^2)^-2/2. Move that back into the denominator (-2/2=-1) to get -6p(p)/(264-2p^2). Factor 2 out, and you get 3p^2/(132-p^2).
Man....taking calc over a winter session is...stressful and chaotic..at best. :P :P
*head asplode*