For the demand function find the elasticity function.
Problem 52.
p=D(x)=18-Sqrt[x]
My answer is [-36x^1/2 + 2x]/x
However, the answer in book is [-2(18-x^1/2)]/[x^1/2]
To find the elasticity function, we need to differentiate the demand function with respect to x and then multiply it by x/p, where p is the price.
Given the demand function: D(x) = 18 - sqrt(x)
Step 1: Differentiate the demand function with respect to x.
dD(x)/dx = -1/2 * (1/sqrt(x)) = -1/(2sqrt(x))
Step 2: Multiply the differentiated demand function by x/p.
[(-1/(2sqrt(x))) * x] / D(x) = (-x)/(2*sqrt(x) * (18 - sqrt(x)))
Simplifying this expression further, we get:
[(-1/2)(x^1/2)] / [(18 - x^1/2) * x^1/2]
You can multiply the numerator and denominator by 2 to get rid of the negative sign in the numerator and 1/2 in the denominator:
[-2(x^1/2)] / [(18 - x^1/2) * x^1/2]
So, the elasticity function is:
[-2(x^1/2)] / [(18 - x^1/2) * x^1/2]
Therefore, the answer you have in your book is correct.