Homework Answers
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,...
(4) Let f(x) (0 if x<0 (a) Show that f is differentiable at z (b) Is...
(4) Let f(x) (0 if x<0 (a) Show that f is differentiable at z (b) Is f'continuous on R? Is f continuous on R? Justify your answer.
For easy reference, f(z)- e- and its derivatives ()-2r(r-1)e 2r(r-1) 4r -8r +2 (x)-e(Az-8r+2)- and (c)...
For easy reference, f(z)- e- and its derivatives ()-2r(r-1)e 2r(r-1) 4r -8r +2 (x)-e(Az-8r+2)- and (c) Find lim (3) What is the horizontal asymptote? (d) Find the local max, local min, and/or inflection points, if they exist. You may use decimals (round to three decimal places) for your answers. (3) (e) Sketch the graph of f. Clearly label or state the points corresponding to the inter- cepts, asymptotes, local maxima and minima, and inflection points (if they exist). (6) 2...
Find the results of next functions 2.-Find the values of a and b such that fis differentiable at x 1 ax+b si 1s. Sol, a...
Find the results of next functions
2.-Find the values of a and b such that fis differentiable at x 1 ax+b si 1s. Sol, a- 2, b-1. f(x)=1si x<1 x-7 si 0<x Sb| f(x) =16/x si x< 3 If ... a) decide a value of b far which f is continuous b)fis differentiable in the value of b that ycu find in part a)? 4.- In the following functions determine what is requested sen(x) si x < mx+b si x...
1. Consider the function -F5 sin(r) for r f(x) =2 for 1< 3 2-25 for 3...
1. Consider the function -F5 sin(r) for r f(x) =2 for 1< 3 2-25 for 3 x2 -9x + 20 Evaluate the following limits You do not have to cite limit laws, but you must show how you arrived at your answer If a limit Does Not Exist, explain why. You should use oo or -oo where applicable Calculating the limit using L'Hopital's Rule will receive NO CREDIT. (a) lim f(x) r-+0 (b) lim f(x)= z-1 (e) lim f(z) (d)...
1. (1 point) Let f(x) = -3 - 9x? +152 +5. Find the open intervals on...
1. (1 point) Let f(x) = -3 - 9x? +152 +5. Find the open intervals on which is increasing (decreasing). Then determine the 2-coordinates of all relative maxima (minima). f is increasing on the intervals f is decreasing on the intervals 3. The relative maxima of foccur at 2 = 4. The relative minima off occur at - 2. Notes: In the first two your answer should either be a single interval, such as (0,1), a comma separated list of...
} and az az 7) Find do and or if z= x + x= rºcoso, y...
} and az az 7) Find do and or if z= x + x= rºcoso, y = rº sing using the chain rule. Write your answer as a function of r and I.
(7) In this problem let X denote the vector space C(0, 1) with the sup norm. (a) Given f e X, define d(f) = f2. : X → X is differentiable, and Prove that φ find φ'(f). (b) Given f e X, define 9(f...
(7) In this problem let X denote the vector space C(0, 1) with the sup norm. (a) Given f e X, define d(f) = f2. : X → X is differentiable, and Prove that φ find φ'(f). (b) Given f e X, define 9(f) = J0 [f(t)]2dt. Prove that Ψ : X → R is differentiable. and find Ψ(f).
(7) In this problem let X denote the vector space C(0, 1) with the sup norm. (a) Given f e X,...
(a) Let Ω = [4, 101 and let A = 16, 6], [8, 10]} 2. (i) Find F(A) (ii) Let X : 2->R be defined by X = 2-1[4,5]-3 . 1 (6,8) Is X, F(A)-measurable? Justify your answer. (b) Let (2, F) be a measurabl...
(a) Let Ω = [4, 101 and let A = 16, 6], [8, 10]} 2. (i) Find F(A) (ii) Let X : 2->R be defined by X = 2-1[4,5]-3 . 1 (6,8) Is X, F(A)-measurable? Justify your answer. (b) Let (2, F) be a measurable space, and let X :2-R. Suppose that X+ is F-measurable. Does this imply that X is F-measurable? Either prove it or give a counterexample.
(a) Let Ω = [4, 101 and let A = 16,...
Let f(x) be the 27-periodic function which is defined by f(x)-cos(x/4) for-π < x < 1. π. (a) Draw the graph of y f(x) over the interval-3π < x < 3π. Is f continuous on R? (b) Find the...
Let f(x) be the 27-periodic function which is defined by f(x)-cos(x/4) for-π < x < 1. π. (a) Draw the graph of y f(x) over the interval-3π < x < 3π. Is f continuous on R? (b) Find the trigonometric Fourier Series (with L π) for f(x). Does the series converge absolutely or conditionally? Does it converge uniformly? Justify your answer. (c) Use your result to obtain explicit values for these three series: 16k2 1 16k2 1 (16k2 1)2 に1...
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,...
(4) Let f(x) (0 if x<0 (a) Show that f is differentiable at z (b) Is f'continuous on R? Is f continuous on R? Justify your answer.
For easy reference, f(z)- e- and its derivatives ()-2r(r-1)e 2r(r-1) 4r -8r +2 (x)-e(Az-8r+2)- and (c) Find lim (3) What is the horizontal asymptote? (d) Find the local max, local min, and/or inflection points, if they exist. You may use decimals (round to three decimal places) for your answers. (3) (e) Sketch the graph of f. Clearly label or state the points corresponding to the inter- cepts, asymptotes, local maxima and minima, and inflection points (if they exist). (6) 2...
Find the results of next functions
2.-Find the values of a and b such that fis differentiable at x 1 ax+b si 1s. Sol, a- 2, b-1. f(x)=1si x<1 x-7 si 0<x Sb| f(x) =16/x si x< 3 If ... a) decide a value of b far which f is continuous b)fis differentiable in the value of b that ycu find in part a)? 4.- In the following functions determine what is requested sen(x) si x < mx+b si x...
1. Consider the function -F5 sin(r) for r f(x) =2 for 1< 3 2-25 for 3 x2 -9x + 20 Evaluate the following limits You do not have to cite limit laws, but you must show how you arrived at your answer If a limit Does Not Exist, explain why. You should use oo or -oo where applicable Calculating the limit using L'Hopital's Rule will receive NO CREDIT. (a) lim f(x) r-+0 (b) lim f(x)= z-1 (e) lim f(z) (d)...
1. (1 point) Let f(x) = -3 - 9x? +152 +5. Find the open intervals on which is increasing (decreasing). Then determine the 2-coordinates of all relative maxima (minima). f is increasing on the intervals f is decreasing on the intervals 3. The relative maxima of foccur at 2 = 4. The relative minima off occur at - 2. Notes: In the first two your answer should either be a single interval, such as (0,1), a comma separated list of...
} and az az 7) Find do and or if z= x + x= rºcoso, y = rº sing using the chain rule. Write your answer as a function of r and I.
(7) In this problem let X denote the vector space C(0, 1) with the sup norm. (a) Given f e X, define d(f) = f2. : X → X is differentiable, and Prove that φ find φ'(f). (b) Given f e X, define 9(f) = J0 [f(t)]2dt. Prove that Ψ : X → R is differentiable. and find Ψ(f).
(7) In this problem let X denote the vector space C(0, 1) with the sup norm. (a) Given f e X,...
(a) Let Ω = [4, 101 and let A = 16, 6], [8, 10]} 2. (i) Find F(A) (ii) Let X : 2->R be defined by X = 2-1[4,5]-3 . 1 (6,8) Is X, F(A)-measurable? Justify your answer. (b) Let (2, F) be a measurable space, and let X :2-R. Suppose that X+ is F-measurable. Does this imply that X is F-measurable? Either prove it or give a counterexample.
(a) Let Ω = [4, 101 and let A = 16,...
Let f(x) be the 27-periodic function which is defined by f(x)-cos(x/4) for-π < x < 1. π. (a) Draw the graph of y f(x) over the interval-3π < x < 3π. Is f continuous on R? (b) Find the trigonometric Fourier Series (with L π) for f(x). Does the series converge absolutely or conditionally? Does it converge uniformly? Justify your answer. (c) Use your result to obtain explicit values for these three series: 16k2 1 16k2 1 (16k2 1)2 に1...
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