A fundamental problem in crystallography is the determination of the packing fraction of a crystal lattice, which is the fraction of space occupied by the atoms in the lattice, assuming that the atoms are hard spheres. When the lattice contains exactly two different kinds of atoms, it can be shown that the packing fraction is given by the formula
f(x) = K(1 + c^2*x^3)
---------------
(1 + x)^3
where x =(r/R) is the ratio of the radii, r and R, of the two kinds of atoms in the lattice, and K and c are positive constants.
(a) The function f(x) has exactly one critical number. Find it and use the second
derivative test to classify it as a relative maximum or a relative minimum.
(b) The numbers c and K and the domain of f(x) depend on the cell structure in the lattice. For ordinary rock salt: c = 1, K = (2π/3), and the domain is the interval
(sqrt(2) − 1) <= x ≤<=1. Find the largest and smallest values of f(x).
(c) Repeat part (b) for β-cristobalite, for which c = sqrt(2), K = sqrt(3π/16), and the domain is 0 <= x <= 1.
(d) What can be said about the packing fraction f(x) if r is much larger than R?
Answer this question by computing lim f(x).
x→∞
Hey! Sorry that I am confused on this question. I don't know where to start. Can someone help me how to work through the problem? I am just lost on what's going on here. I think I am that type of person who likes to work backwards to see how to do this problem.
Thank you in advance! :)
(a) well, you know critical numbers are where f'(x) is zero or undefined.
we can factor out the K and ignore it, since it is just a scale factor. Using
f(x) = (1+c^2x^3)/(1+x)^3
f' =
(3c^2x^2)(1+x)^3 - (1+c^2x^3)(3)(1+x)^2
----------------------------------------------
(1+x)^6
or, factoring out the (a+x)^2,
(1+c^2x^3)/(1+x)^3
f' =
(3c^2x^2)(1+x) - 3(1+c^2x^3)
---------------------------------
(1+x)^4
f'(x) is undefined at x = -1
f'(x)=0 when the numerator is zero. That is, when
(3c^2x^2)(1+x) - 3(1+c^2x^3) = 0
3(cx-1)(cx+1) = 0
x = ±1/c
Maybe this will get you started. Gotta hit the rack now, but I'll check back in in a few hours.
I'm working on the same problem, any chance you could help me through the second derivative test section?
No problem! Let's break down the problem step by step and work through it together.
(a) To find the critical number of the function f(x), we need to find the value of x where the derivative of f(x) equals zero.
1. Start by taking the derivative of f(x) with respect to x:
f'(x) = K(1 + c^2*x^3) * (3*c^2*x^2) * (1 + x)^3 - K(1 + c^2*x^3) * 3*(1 + x)^2
2. Set f'(x) equal to zero and solve for x:
0 = K(1 + c^2*x^3) * (3*c^2*x^2) * (1 + x)^3 - K(1 + c^2*x^3) * 3*(1 + x)^2
Now, you can factor out the common terms and simplify the equation. You should end up with an equation in terms of x. Solve this equation to find the critical number x.
Once you have the critical number x, you can use the second derivative test to classify it as a relative maximum or a relative minimum. The second derivative test involves evaluating the second derivative of f(x) at the critical number and checking its sign. If the second derivative is positive, it's a relative minimum. If the second derivative is negative, it's a relative maximum. If the second derivative is zero, then the test is inconclusive.
(b) Now let's move on to the next part of the problem - finding the largest and smallest values of f(x) for the given values of c, K, and the domain.
Plug in the values of c = 1, K = (2π/3), and the domain (sqrt(2) − 1) <= x <= 1 into the formula for f(x). Calculate f(x) for the lower bound of the domain and the upper bound of the domain. Compare the results to find the largest and smallest values of f(x).
(c) Repeat the same process as part (b) but now use the values c = sqrt(2), K = sqrt(3π/16), and the domain 0 <= x <= 1 for β-cristobalite.
(d) The last part of the problem asks us to find the limit of f(x) as x approaches infinity (lim f(x) as x→∞).
To evaluate this limit, substitute infinity into the formula for f(x) and see what it approaches. You may have to consider the behavior of the terms in the formula as x gets larger.
I hope this breakdown helps! If you have any more specific questions or need further assistance, feel free to ask.