What is the upper envelope of the family of ballistic curves?

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In summary, the upper envelope for the family of ballistic curves is y = 1/2 - x^2/2. To find this, one must find the maximum of g_x(a) = ax - [x^2(a^2+1)]/2, which occurs at a=1/x. Then, substituting this value back into the original equation, we get y=F(x)=(1/2) x^2.
  • #1
mathmathmad
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Homework Statement


find the upper envelope for the family of ballistic curves :

y = ax - [x^2(a^2+1)]/2

an upper envelope is a curve y=F(x) such that for each x fixed, F(x) is the maximum of g(a) = ax - [x^2(a^2+1)]/2 for a in R

Homework Equations


The Attempt at a Solution


diff g(a) wrt to a and equate to 0 and get a=1/x so F(x) is 1/x?? :(
 
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  • #2
You are right to start by finding the maximum of [tex]g_x(a) = ax - \frac12 x^2 (a^2 + 1)[/tex], but you need to prove that the maximum of [tex]g_x[/tex] occurs at the single critical point you find.

However, you have found the value of [tex]a[/tex] which maximizes [tex]g_x(a) = F(x)[/tex] at that [tex]x[/tex] --- not what you want, which is the value of the function at that point, expressed as a function of [tex]x[/tex]. What further computation is necessary here?
 
  • #3
Am I right in thinking you then have to sub the a=1/x back into the original y to get,

y=F(x)=(1/2) x^2

and mathmathmad your a warwick student right?
 
  • #4
blackscorpion said:
Am I right in thinking you then have to sub the a=1/x back into the original y to get,

y=F(x)=(1/2) x^2

and mathmathmad your a warwick student right?

I do not think so as it is not related with F(x) being maximum of g(a). Btw blackscorpian, I've sent you a pm.
 
  • #5
is it F(x)=1/2 - x^2/2 instead?

@blackscorpion : yes... I am. you too?
 
  • #6
Yes it is
 
  • #7
Ahhh crap, it is aswell.
Stupidly canceled the ones forgettin bout the over 2 part.
Well that's a mark thrown away, lol
 

1. What is the upper envelope of a function?

The upper envelope of a function is the maximum value of the function over a given domain. It represents the highest point that the function reaches within that domain.

2. How is the upper envelope of a function useful?

The upper envelope can provide important information about the behavior and trends of a function. It can help identify the highest values of the function and any potential outliers.

3. How is the upper envelope different from the maximum value of a function?

The maximum value of a function is the single highest point that the function reaches, while the upper envelope represents the highest points of the function over a given domain. This means that the maximum value is a specific point, while the upper envelope is a curve or line.

4. Can the upper envelope of a function be easily determined?

The upper envelope of a function can be determined through various methods such as graphing, calculating derivatives, or using mathematical software. However, for complex functions, it may be difficult to determine the exact upper envelope and approximation methods may be necessary.

5. How is the upper envelope of a function related to the lower envelope?

The upper envelope and lower envelope of a function are closely related as they both represent the extreme values of the function over a given domain. The upper envelope is the highest points of the function while the lower envelope is the lowest points.

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