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 (D(x)) with respect to x and then multiply it by x/D(x). Let's go step by step to determine if your answer or the answer in the book is correct.
Starting with the demand function:
D(x) = 18 - sqrt(x)
To find the elasticity function, we first differentiate the demand function with respect to x:
D'(x) = d(18 - sqrt(x))/dx
Using the power rule of differentiation, we get:
D'(x) = 0 - (1/2)x^(-1/2)
Simplifying further:
D'(x) = -1/(2*sqrt(x))
Now, we need to multiply this derivative by x/D(x):
E(x) = D'(x) * (x / D(x))
Plugging in the values of D(x) and D'(x), we have:
E(x) = (-1/(2*sqrt(x))) * (x / (18 - sqrt(x)))
Now let's simplify your answer and the answer given in the book to see which one is correct.
Your answer: [-36x^1/2 + 2x] / x
To simplify it further, we can separate the terms inside the square brackets:
[-36x^1/2/x + 2x/x]
Simplifying and canceling out common terms:
[-36/x^1/2 + 2]
Now, let's simplify the answer given in the book:
[-2(18 - x^1/2)] / [x^1/2]
Multiplying the -2 through to simplify, we have:
[(-2 * 18) - (-2 * x^1/2)] / [x^1/2]
[-36 + 2x^1/2] / [x^1/2]
Comparing your answer with the answer in the book, we see that they are equivalent. Your answer and the one given in the book are the same, just written in slightly different forms.
Therefore, both answers [-36x^1/2 + 2x]/x and [-2(18 - x^1/2)]/[x^1/2] are correct and represent the elasticity function for the given demand function.