Find Price Elasticity of Demand

[mathkid3]

Find price elasticity of demand for the demand function at the indicated x-value.

Is the demand elastic, inelastic or of unit elasticity at the indicated x-value?

Use a graphing utility to graph the revenue function, and identify the intervals of elasticity and inelasticity. ... I will list all information I can to help someone to help me to solve and understand the problem.

Demand Function = p = (500)/(x^2) + 5

x-value = 5

The following may be of help as well...

Definition of Price Elasticity of Demand - If p = f(x) is a differentiable function, then the price elasticity of demand is given by

N = (p/x)/(dp/dx)

where N is the lowercase Greek letter eta. For a given price, the demand is elastic when abs(absolute) value N > 1, the demand is inelastic when abs N < 1, and the demand has unit elasticity when abs N = 1Thanks in advance for any and all assistance on this Calculus problem

-mathkid3 (Happy)

 

[MarkFL]

Well, I would begin by computing $\displaystyle \frac{dp}{dx}|_{x=5}$

Next, I would compute $\displaystyle \frac{p(5)}{5}$.

Once you have these two numbers, then compute their quotient to get the price elasticity of demand $\displaystyle \eta$.

Now, let's see what you get...

By the way, this site also supports the rendering of $\displaystyle \LaTeX$. The difference is (at least the way I do it) is to enclose your code with the tags:

$\displaystyle insert code here$

 

[mathkid3]

Mark,

so glad you are on this site and you have already helped me so much with understanding some of these Calculus problems.

ok ..let's see...

You said first to find the derV of p so I get

(x-1000)/(x^3)

next you say find p(5)/(5)

Areyou saying to plug 5 into the original function and then divided the outcome by 5?

if so I get x = 5

am I correct thus far and if so tell me again what to do next?

If I am not correct, would you explain to me again what to do ?

Thank You

 

[CaptainBlack]

The following may be of help as well...

Definition of Price Elasticity of Demand - If p = f(x) is a differentiable function, then the price elasticity of demand is given by

N = (p/x)/(dp/dx)

where N is the lowercase Greek letter eta. For a given price, the demand is elastic when abs(absolute) value N > 1, the demand is inelastic when abs N < 1, and the demand has unit elasticity when abs N = 1

Your definition of elasticity:
\[E_d=\frac{p}{x \frac{dp}{dx}}\]
is the reciprical of that given on the Wikipedia page linked to in your other thread:
\[E_d=\frac{x}{p}\frac{dp}{dx}\]
(Note that the Wikipedia page uses \(Q\) for demand (quantity) and \(P\) for price).

Now the table the Wikipedia page gives for interpreting \(E_p\) is exactly the same as that given above by you, so there is a mistake somewhere.

CB

 

[CellCoree]

I am happy to assist you with this problem. To find the price elasticity of demand, we will use the formula N = (p/x)/(dp/dx) as given in the definition.
First, we need to find the derivative of the demand function with respect to x.
dp/dx = -1000/x^3
Now, we can plug in the given x-value of 5 into both the demand function and the derivative to find the corresponding values.
p = (500)/(5^2) + 5 = 105
dp/dx = -1000/(5^3) = -8
Now, we can plug these values into the price elasticity of demand formula to find the value of N at x=5.
N = (105/5)/(-8) = -21/8
Since the absolute value of N is greater than 1, we can conclude that the demand is elastic at x=5.
To graph the revenue function, we can use a graphing utility and plot the function R(x) = x*p(x) where p(x) is the demand function given.
We can see that the graph has two intervals of elasticity, one on the left side of x=5 and one on the right side. The interval of inelasticity is between these two intervals of elasticity.
I hope this helps you understand the problem better. Let me know if you need any further clarification.