Newton's Method/Trapezoidal Rule

[johnhuntsman]

My professor had asked that I solve the follwing problem using Newton's Method and the Trapezoidal Rule:

"A company modeled the demand curve for its product (in dollars) by the equation

p = (800000e^(- x / 5000)) / (x + 20000).

Use Newton's Method to estimate the sales level (x I'm pretty sure) when the selling price is $16. Then find the approximate consumer surplus for this level."

I'm currently stuck on how Newton's Method is meant to find x when p = 16.

The derivative of p is:

- (160e^(- x / 5000) * (x + 25000)) / (x + 20000) ^ 2

 

[Chestermiller]

My professor had asked that I solve the follwing problem using Newton's Method and the Trapezoidal Rule:

"A company modeled the demand curve for its product (in dollars) by the equation

p = (800000e^(- x / 5000)) / (x + 20000).

Use Newton's Method to estimate the sales level (x I'm pretty sure) when the selling price is $16. Then find the approximate consumer surplus for this level."

I'm currently stuck on how Newton's Method is meant to find x when p = 16.

The derivative of p is:

- (160e^(- x / 5000) * (x + 25000)) / (x + 20000) ^ 2

To get Newton's Method started, you need an initial guess for x.
What is the value of p when x is zero? If this is close to 16, this value of x might be a good initial guess for Newton's method. What is the initial error in p for this value of x? What is the Newton Method formula?

 

[johnhuntsman]

To get Newton's Method started, you need an initial guess for x.
What is the value of p when x is zero? If this is close to 16, this value of x might be a good initial guess for Newton's method. What is the initial error in p for this value of x? What is the Newton Method formula?

x is never equal to zero. That's the thing. Unless I'm mistaken.

http://www.wolframalpha.com/input/?i=y+=+(800000e^(-x/5000))/(x+20000)

 

[Chestermiller]

x is never equal to zero. That's the thing. Unless I'm mistaken.

http://www.wolframalpha.com/input/?i=y+=+(800000e^(-x/5000))/(x+20000)

That doesn't matter. I repeat my question. What is the value of p (in the equation) if x is set equal to zero?

 

[johnhuntsman]

p = 40 when x = 0

 

[Chestermiller]

p = 40 when x = 0

Good. Now, the problem you are trying to solve is:

f(x) = (800000e^(- x / 5000)) / (x + 20000) -16 =0

Your initial guess is x =0, and at x = 0, f(x) = 40 -16 = 34

Your Newton method equation is:

x_n+1 = x_n-f(x_n)/f '(x_n)

with x₁ =0 and f(x₁)= 34

Now find f ' (x₁) , and then x₂

If that's not close enough to satisfy f(x) = 0, do another iteration.

 

[johnhuntsman]

I see now what to do now. Thanks bunches : D

 

[Ray Vickson]

My professor had asked that I solve the follwing problem using Newton's Method and the Trapezoidal Rule:

"A company modeled the demand curve for its product (in dollars) by the equation

p = (800000e^(- x / 5000)) / (x + 20000).

Use Newton's Method to estimate the sales level (x I'm pretty sure) when the selling price is $16. Then find the approximate consumer surplus for this level."

I'm currently stuck on how Newton's Method is meant to find x when p = 16.

The derivative of p is:

- (160e^(- x / 5000) * (x + 25000)) / (x + 20000) ^ 2

In problems of this type it is always a good idea to "scale" the problem properly (even if you use a computer!). Instead of x it would be better to use, say y = x/5000, so your equation becomes p = 800,000 exp(-y)/(5000 y + 20,000) = 160*exp(-y)/(y+4).

Now, for p = 16 you need to solve 10*exp(-y) = y+4, or 10*exp(y)-y-4 = 0. Using f(y) = 10*exp(-y) - y - 4 instead of 160*exp(-y)/(y+4)-16 is a lot easier in Newton's method, since the derivative is a lot easier.

RGV