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3. (pp. 592, Marsden & Hoffman] Suppose that f:1-1, 1] → R is differentiable on (–,],...
1. (pp. 592, Marsden & Hoffman) For each of the functions determine what type of convergence...
1. (pp. 592, Marsden & Hoffman) For each of the functions determine what type of convergence the Fourier series will have and if we can differentiate the series. b) f (x) = T – x2 on (-7,].
2. [pp. 492, Marsden & Hoffman] Let p [a, bR and :R R be continuous. Show that R} C R2 /-{(z, p (...
2. [pp. 492, Marsden & Hoffman] Let p [a, bR and :R R be continuous. Show that R} C R2 /-{(z, p (x)) : x e [a, b]} C R2 has volume zero in R2 and the set B-{(x, ψ(x)) : x has measure zero in R ie set F
2. [pp. 492, Marsden & Hoffman] Let p [a, bR and :R R be continuous. Show that R} C R2 /-{(z, p (x)) : x e [a, b]} C R2...
2. Let f: R R be a continuous function. Suppose that f is differentiable on R\{0} and that there exists an L e R such that lim,of,(z) = L. Prove that f is differentiable at 1-0 with f,(0) = L. (Hint:...
2. Let f: R R be a continuous function. Suppose that f is differentiable on R\{0} and that there exists an L e R such that lim,of,(z) = L. Prove that f is differentiable at 1-0 with f,(0) = L. (Hint: Use the definition of derivative and then use mean value theorem)
2. Let f: R R be a continuous function. Suppose that f is differentiable on R\{0} and that there exists an L e R such that lim,of,(z) =...
Now suppose f' is continuous and f" is piecewise continuous on (0, L). (b) If f(0)...
Now suppose f' is continuous and f" is piecewise continuous on (0, L). (b) If f(0) = f(L) = 0, then O f(x) = į bn sin ηπα L 0<x<L, n=1 612 Chapter 11 Boundary Value Problems and Fourier Expansions with bn = 2L n272 S“ r"(a)sin пах dr. L (11.3.5) Solve the initial-boundary value problem. Theorem 11.3.5 (b) will simplify the computa- tion of the coefficients in the Fourier sine series. Uit = 64uze, 0<r <3, t > 0,...
(a) Suppose f is continuously differentiable on the closed and bounded interval I = [0, 1]....
(a) Suppose f is continuously differentiable on the closed and bounded interval I = [0, 1]. Show that f is uniformly continuous on I. (b) Suppose g is continuously differentiable on the open interval J = (0,1). Give and example of such a function which is NOT uniformly continuous on J, and prove your answer.
Exercise 1. Let f : R R be differentiable on la, b, where a, b R...
Exercise 1. Let f : R R be differentiable on la, b, where a, b R and a < b, and let f be continuous on [a, b]. Show that for every e> 0 there exists a 6 > 0 such that the inequality f(x)- f(c) T-C holds for all c, x E [a, 히 satisfying 0 < |c-x| < δ
2. Let f R R and g R-R be functions that are continuous on1,1 and differentiable on (1,1). Suppose that f(-1-f(1) and 9(-1). Show that there exists c e (1,1) such that 2. Let f R R and g R-R...
2. Let f R R and g R-R be functions that are continuous on1,1 and differentiable on (1,1). Suppose that f(-1-f(1) and 9(-1). Show that there exists c e (1,1) such that
2. Let f R R and g R-R be functions that are continuous on1,1 and differentiable on (1,1). Suppose that f(-1-f(1) and 9(-1). Show that there exists c e (1,1) such that
1) Suppose f (a, b) R is continuous. The Carathéodory Theorem says that f(x) is differentiable at -cE (a, b) if 3 (a, b)-R which is continuous, and so that, (a) Show, for any constant a and conti...
1) Suppose f (a, b) R is continuous. The Carathéodory Theorem says that f(x) is differentiable at -cE (a, b) if 3 (a, b)-R which is continuous, and so that, (a) Show, for any constant a and continuous function (x), that af(x) is continuous at z-c by finding a Carathéodory function Paf(x). (b) Show, for any constants a, B, that if g : (a, b) -R is differentiable at c, with Carathéodory function pg(z), then the linear combination of functions,...
3. (a) Suppose f : (a, b) + R is differentiable, and there exists M E...
3. (a) Suppose f : (a, b) + R is differentiable, and there exists M E R such that If'(x) < M for all x € (a, b). Prove that f is uniformly continuous on (a, b). (b) Let f : [0, 1] → [0, 1] be a continuous function. Prove that there exists a point pe [0, 1] with f(p) = p.
A function is defined over (0,3) by f(3) = 12 +1. We then extend it to...
A function is defined over (0,3) by f(3) = 12 +1. We then extend it to an even periodic function of period 6 and its graph is displayed below. 2 15 0.5 5 10 15 х -0.5 The function may be approximated by the Fourier series f () = ap + 01 (an cos ( 122 ) + bn sin (022)). where L is the half-period of the function. Use the fact that f(x) sin is an odd functions, enter...
1. (pp. 592, Marsden & Hoffman) For each of the functions determine what type of convergence the Fourier series will have and if we can differentiate the series. b) f (x) = T – x2 on (-7,].
2. [pp. 492, Marsden & Hoffman] Let p [a, bR and :R R be continuous. Show that R} C R2 /-{(z, p (x)) : x e [a, b]} C R2 has volume zero in R2 and the set B-{(x, ψ(x)) : x has measure zero in R ie set F
2. [pp. 492, Marsden & Hoffman] Let p [a, bR and :R R be continuous. Show that R} C R2 /-{(z, p (x)) : x e [a, b]} C R2...
2. Let f: R R be a continuous function. Suppose that f is differentiable on R\{0} and that there exists an L e R such that lim,of,(z) = L. Prove that f is differentiable at 1-0 with f,(0) = L. (Hint: Use the definition of derivative and then use mean value theorem)
2. Let f: R R be a continuous function. Suppose that f is differentiable on R\{0} and that there exists an L e R such that lim,of,(z) =...
Now suppose f' is continuous and f" is piecewise continuous on (0, L). (b) If f(0) = f(L) = 0, then O f(x) = į bn sin ηπα L 0<x<L, n=1 612 Chapter 11 Boundary Value Problems and Fourier Expansions with bn = 2L n272 S“ r"(a)sin пах dr. L (11.3.5) Solve the initial-boundary value problem. Theorem 11.3.5 (b) will simplify the computa- tion of the coefficients in the Fourier sine series. Uit = 64uze, 0<r <3, t > 0,...
(a) Suppose f is continuously differentiable on the closed and bounded interval I = [0, 1]. Show that f is uniformly continuous on I. (b) Suppose g is continuously differentiable on the open interval J = (0,1). Give and example of such a function which is NOT uniformly continuous on J, and prove your answer.
Exercise 1. Let f : R R be differentiable on la, b, where a, b R and a < b, and let f be continuous on [a, b]. Show that for every e> 0 there exists a 6 > 0 such that the inequality f(x)- f(c) T-C holds for all c, x E [a, 히 satisfying 0 < |c-x| < δ
2. Let f R R and g R-R be functions that are continuous on1,1 and differentiable on (1,1). Suppose that f(-1-f(1) and 9(-1). Show that there exists c e (1,1) such that
2. Let f R R and g R-R be functions that are continuous on1,1 and differentiable on (1,1). Suppose that f(-1-f(1) and 9(-1). Show that there exists c e (1,1) such that
1) Suppose f (a, b) R is continuous. The Carathéodory Theorem says that f(x) is differentiable at -cE (a, b) if 3 (a, b)-R which is continuous, and so that, (a) Show, for any constant a and continuous function (x), that af(x) is continuous at z-c by finding a Carathéodory function Paf(x). (b) Show, for any constants a, B, that if g : (a, b) -R is differentiable at c, with Carathéodory function pg(z), then the linear combination of functions,...
3. (a) Suppose f : (a, b) + R is differentiable, and there exists M E R such that If'(x) < M for all x € (a, b). Prove that f is uniformly continuous on (a, b). (b) Let f : [0, 1] → [0, 1] be a continuous function. Prove that there exists a point pe [0, 1] with f(p) = p.
A function is defined over (0,3) by f(3) = 12 +1. We then extend it to an even periodic function of period 6 and its graph is displayed below. 2 15 0.5 5 10 15 х -0.5 The function may be approximated by the Fourier series f () = ap + 01 (an cos ( 122 ) + bn sin (022)). where L is the half-period of the function. Use the fact that f(x) sin is an odd functions, enter...
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