Homework Answers
![1) Brillouni function can be expressed as follows 2J +1 2J 2J +1 2J 2J 2J +1 2J+1 Using the following , expansion series coth (x)- -+1-+ x 3 45 -coth 2J AsJ->oo --coth 2J Therefore As J -00 B, (a)-coth[a]--= L(a) 1 2J+1 2 2J B.(a)-2coth (2a)-coth (a) (2) As J →-, → 2,implies J →-, (by simplifying the above equation = tanh (a) 2/+1 α 2J -2 27,-77e+(2.1+1) 3(2) 2J Therefore B,(a) 27 a(21+1) 2J α 3J](http://img.homeworklib.com/questions/b4612d90-3a7e-11eb-8927-0dcc7a04fd01.png?x-oss-process=image/resize,w_560)
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Show that the Brillouin function approaches the Langevin function as J-+ oo. What are the limits...
1) Use the definition of continuity and the property of limits to show that the function...
1) Use the definition of continuity and the property of limits to show that the function is continuous at x² + 5x the given number a. a = 2 2x+1
Limits at Ifinity and Limits of Sequences 787 57. Business The cost function for a certain model ...
Limits at Ifinity and Limits of Sequences 787 57. Business The cost function for a certain model of a personal digital assistant (PDA) is given by C I 3.50x + 45,750, where C is the cost (in dollars) and x is the number of PDAs produced. (a) Write a model for the average cost per unit produced (b) Find the average costs per unit when 100 and x 1000. (c) Determine the limit of the average cost function as x...
(d) The function f(x)1 is locally integrable on (0, oo). To see whether converges, we consider th...
(d) The function f(x)1 is locally integrable on (0, oo). To see whether converges, we consider the improper integrals separately. (The choice of π above is arbitrary.) By considering f (x) lim an show that 11 converges iff p< 1. Next, by considering lim J(z) an -p- dx show that /2 converges iff p +q>1. Finally, combine these results to show that I converges iff p < 1 and p+q1.
(d) The function f(x)1 is locally integrable on (0, oo)....
Answer C 6. Let f be a continuous function on [0, oo) such that 0 f(z)...
Answer C
6. Let f be a continuous function on [0, oo) such that 0 f(z) Cl- for some C,e> 0, and let a = fo° f(x) da. (The estimate on f implies the convergence of this integral.) Let fk(x) = kf(ka) a. Show that lim00 fk(x) = 0 for all r > 0 and that the convergence is uniform on [8, oo) for any 6> 0. b. Show that limk00 So ()dz = a. c. Show that lim00 So...
im having trouble understanding limits (1) Use the given graph of the function f(x) to answer...
im having trouble understanding limits
(1) Use the given graph of the function f(x) to answer the following questions. 3 2 0 0 -1 -2 Find the limit of f(x) as x -+0+ for ” does not exist” enter ” DNE” 79 79 99 for enter negative in finity” for ” +00” enter " positive in finity" (1) Use the given graph of the function f(x) to answer the following questions. 3 2 0 0 1 2 3 4 -27...
2. Let (En)acy be a sequence in R (a) Show that xn → oo if and only if-An →-oo. (b) If xn > 0 for...
#s 2, 3, 6
2. Let (En)acy be a sequence in R (a) Show that xn → oo if and only if-An →-oo. (b) If xn > 0 for all n in N, show that linnAn = 0 if and only if lim-= oo. 3. Let ()nEN be a sequence in R. (a) If x <0 for all n in N, show that - -oo if and only if xl 0o. (b) Show, by example, that if kal → oo,...
16. Epsilon-Delta Limits (15pt]: (a) Let c be an interior point in the interval (a,b) over which the function I is defined ve the "epsilon-delta" definition of what it means for f(a) to h...
16. Epsilon-Delta Limits (15pt]: (a) Let c be an interior point in the interval (a,b) over which the function I is defined ve the "epsilon-delta" definition of what it means for f(a) to have a limit L, a approaches c. b) Consider the function (C) -", which has the limit lif.,Sind largest value for δ that satisfies the limit definition at this point.
16. Epsilon-Delta Limits (15pt]: (a) Let c be an interior point in the interval (a,b) over which...
4. (a) Let A [0, oo) and let f.g:AR be functions which are continuous at 0 and are such that f(0) 9(0)-1. Show that there exists some δ > 0 such that ifTE 0,d) then (b) Consider the function 0 l i...
4. (a) Let A [0, oo) and let f.g:AR be functions which are continuous at 0 and are such that f(0) 9(0)-1. Show that there exists some δ > 0 such that ifTE 0,d) then (b) Consider the function 0 l if z e R is rational, if zER is irrational f(z) Show that limfr) does not exists for any ceR.
4. (a) Let A [0, oo) and let f.g:AR be functions which are continuous at 0 and are such...
(2) Let X be a locally compact Hausdorff space, and let μ be a regular Borel measure on X such that μ(X) = +oo. Show that there is a non-negative function f CO(X) such that Jfdlı-+oo. Idea. Construct...
(2) Let X be a locally compact Hausdorff space, and let μ be a regular Borel measure on X such that μ(X) = +oo. Show that there is a non-negative function f CO(X) such that Jfdlı-+oo. Idea. Construct a sequence {K f-Σ001 nzfn, n} of disjoint compact sets K n with μ(An) > n and set where fn E Co(X) with XKn S f 31 く!
(2) Let X be a locally compact Hausdorff space, and let μ be a...
Compute the Laplace transform of the function on [0, oo). Here, uc (t)ut c where u(t)...
Compute the Laplace transform of the function on [0, oo). Here, uc (t)ut c where u(t) is the Heaviside function on [0, oo). Give your answer as a function of 8 for 8 〉 0. (f)(s) =
1) Use the definition of continuity and the property of limits to show that the function is continuous at x² + 5x the given number a. a = 2 2x+1
Limits at Ifinity and Limits of Sequences 787 57. Business The cost function for a certain model of a personal digital assistant (PDA) is given by C I 3.50x + 45,750, where C is the cost (in dollars) and x is the number of PDAs produced. (a) Write a model for the average cost per unit produced (b) Find the average costs per unit when 100 and x 1000. (c) Determine the limit of the average cost function as x...
(d) The function f(x)1 is locally integrable on (0, oo). To see whether converges, we consider the improper integrals separately. (The choice of π above is arbitrary.) By considering f (x) lim an show that 11 converges iff p< 1. Next, by considering lim J(z) an -p- dx show that /2 converges iff p +q>1. Finally, combine these results to show that I converges iff p < 1 and p+q1.
(d) The function f(x)1 is locally integrable on (0, oo)....
Answer C
6. Let f be a continuous function on [0, oo) such that 0 f(z) Cl- for some C,e> 0, and let a = fo° f(x) da. (The estimate on f implies the convergence of this integral.) Let fk(x) = kf(ka) a. Show that lim00 fk(x) = 0 for all r > 0 and that the convergence is uniform on [8, oo) for any 6> 0. b. Show that limk00 So ()dz = a. c. Show that lim00 So...
im having trouble understanding limits
(1) Use the given graph of the function f(x) to answer the following questions. 3 2 0 0 -1 -2 Find the limit of f(x) as x -+0+ for ” does not exist” enter ” DNE” 79 79 99 for enter negative in finity” for ” +00” enter " positive in finity" (1) Use the given graph of the function f(x) to answer the following questions. 3 2 0 0 1 2 3 4 -27...
#s 2, 3, 6
2. Let (En)acy be a sequence in R (a) Show that xn → oo if and only if-An →-oo. (b) If xn > 0 for all n in N, show that linnAn = 0 if and only if lim-= oo. 3. Let ()nEN be a sequence in R. (a) If x <0 for all n in N, show that - -oo if and only if xl 0o. (b) Show, by example, that if kal → oo,...
16. Epsilon-Delta Limits (15pt]: (a) Let c be an interior point in the interval (a,b) over which the function I is defined ve the "epsilon-delta" definition of what it means for f(a) to have a limit L, a approaches c. b) Consider the function (C) -", which has the limit lif.,Sind largest value for δ that satisfies the limit definition at this point.
16. Epsilon-Delta Limits (15pt]: (a) Let c be an interior point in the interval (a,b) over which...
4. (a) Let A [0, oo) and let f.g:AR be functions which are continuous at 0 and are such that f(0) 9(0)-1. Show that there exists some δ > 0 such that ifTE 0,d) then (b) Consider the function 0 l if z e R is rational, if zER is irrational f(z) Show that limfr) does not exists for any ceR.
4. (a) Let A [0, oo) and let f.g:AR be functions which are continuous at 0 and are such...
(2) Let X be a locally compact Hausdorff space, and let μ be a regular Borel measure on X such that μ(X) = +oo. Show that there is a non-negative function f CO(X) such that Jfdlı-+oo. Idea. Construct a sequence {K f-Σ001 nzfn, n} of disjoint compact sets K n with μ(An) > n and set where fn E Co(X) with XKn S f 31 く!
(2) Let X be a locally compact Hausdorff space, and let μ be a...
Compute the Laplace transform of the function on [0, oo). Here, uc (t)ut c where u(t) is the Heaviside function on [0, oo). Give your answer as a function of 8 for 8 〉 0. (f)(s) =
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