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Implicit Function Theorem in Two Variables: Let g: R2 → R be a smooth function. Set {(z, y) E R2 ...
Implicit Function Theorem in Two Variables: Let g: R2 → R be a smooth function. Set {(z, y) E R2 | g(z, y) = 0} S Suppose g(a, b)-0 so that (a, b) E S and dg(a, b)メO. Then there exists an open neighborhood of (a, b) say V such that SnV is the image of a smooth parameterized curve. (1) Verify the implicit function theorem using the two examples above. 2) Since dg(a,b) 0, argue that it suffices to...
5. Let f,g:R + R be continuous 27-periodic functions. Define h: R + R by 271...
5. Let f,g:R + R be continuous 27-periodic functions. Define h: R + R by 271 h(s) = 5" (3 – t)g(t) dt. Prove that 27 /** h(s) ds (* sc) d) ($* $11) t). 6. Let f : R2 + R be a C2 function. Use Fubini's theorem to show that д?f д?f дхду дудх
2. Let E CRn+m. For every x ER", let Ex := {y € R™ st. (,y)...
2. Let E CRn+m. For every x ER", let Ex := {y € R™ st. (,y) € E}. Let fel'(E). Proved that • for a.e. 2 ER", the function Ex = y + f(x,y) belongs to L'(Ex), • the function R"3o-. s(z. y) dy belongs to L (RM), • we have that sss=fen (562, )dy) ds. This is a slightly stronger version of the Fubini's Theorem that we proved in class, for instance one could define fxs and use the...
4. Define the function f: 0,00) +R by the formula f(x) = dt. +1 Comment: The...
4. Define the function f: 0,00) +R by the formula f(x) = dt. +1 Comment: The integrand does not have a closed form anti-derivative, so do not try to answer the following questions by computing an anti-derivative. Use some properties that we learned. (a) (4 points). Prove that f(x) > 0 for all x > 0, hence f: (0,00) + (0,0). (b) (4 points). Prove that f is injective. (c) (6 points). Prove that f: (0,00) (0,00) is not surjective,...
3. (a) Let f be an infinitely differentiable function on R and define х F(x) =...
3. (a) Let f be an infinitely differentiable function on R and define х F(x) = e-y f(y) dy. Find and prove a formula for F(n), the nth derivative of F. (b) Show that if f is a polynomial then there exists a constant C such that F(n)(x) = Cem for sufficiently large n. Find the least n for which it is true.
10. (10 points) A function f : R2 + R is called a probability density function...
10. (10 points) A function f : R2 + R is called a probability density function on D CR if (6) f(, y) 0 for all (x, y) E D and (i) SD. f(x,y)dA= 1. ſk(1 – 22 – y2) 22 + y2 <1 (a) For what constant k is the function f(z,y) a prob- 12 + y2 > 1 ability density function? Note that D= {(1, Y) ER? : x2 + y² <1}, the closed unit disk in R2...
Problem 3. Define the function: 2+_ 0 if (z,y)#10.0) if (a,y)-(0,0) f(x, v)= (a) Graph the...
Problem 3. Define the function: 2+_ 0 if (z,y)#10.0) if (a,y)-(0,0) f(x, v)= (a) Graph the top portion of the function using Geogebra. Does the function appear to be continuus at 0? (b) Find fz(z, y) and fy(z, y) when (z, y) #10.0) (c) Find f(0,0) and s,(0,0) using the limit definitions of partial derivatives and f,(0,0)-lim rah) - f(O,0) d) Use these limit definitions to show that fay(0,0)--1, while x(0,0)-1 (e) Can we conclude from Clairaut's theorem that()-yr(x,y) for...
Please answer this question Implicit Function Theorem in Two Variables: Let g: R2 - R be...
Please answer this question
Implicit Function Theorem in Two Variables: Let g: R2 - R be a smooth function. Set Suppose g(a, b)-0 so that (a, b) є S and dg(a, b) 0. Then there exists an open neighborhood of (a, b) say V such that SnV is the image of a smooth parameterized curve. (1) Verify the implicit function theorem using the two examples above (2) Since dg(a, b)メ0, argue that it suffices to assume a,b)メ0. (3) Prove the...
4. f(x) is a continuous function on [0, 1] and So f (x)dx = a, where...
4. f(x) is a continuous function on [0, 1] and So f (x)dx = a, where a is constant. Evaluate the following double integral f(x)f(y)dydx. (Hint: Change the order of the integration and use the property of the double integral, so that you can apply Fubini's theorem.)
Consider f : [0, 1] x [0, 1] C R2 + R defined by f(x,y) =...
Consider f : [0, 1] x [0, 1] C R2 + R defined by f(x,y) = ſi if y is rational 2x if y is irrational Show that f is not integrable over R by the following steps: in (a) For each n > 1, find a Sn:= Eosi,jan f(a 6? b., in [0, 1] for 0 < i, j < n such that the Riemann sum converges as n + 0.[10 pts] n 1 n2 n i, ja (b)...
Implicit Function Theorem in Two Variables: Let g: R2 → R be a smooth function. Set {(z, y) E R2 | g(z, y) = 0} S Suppose g(a, b)-0 so that (a, b) E S and dg(a, b)メO. Then there exists an open neighborhood of (a, b) say V such that SnV is the image of a smooth parameterized curve. (1) Verify the implicit function theorem using the two examples above. 2) Since dg(a,b) 0, argue that it suffices to...
5. Let f,g:R + R be continuous 27-periodic functions. Define h: R + R by 271 h(s) = 5" (3 – t)g(t) dt. Prove that 27 /** h(s) ds (* sc) d) ($* $11) t). 6. Let f : R2 + R be a C2 function. Use Fubini's theorem to show that д?f д?f дхду дудх
2. Let E CRn+m. For every x ER", let Ex := {y € R™ st. (,y) € E}. Let fel'(E). Proved that • for a.e. 2 ER", the function Ex = y + f(x,y) belongs to L'(Ex), • the function R"3o-. s(z. y) dy belongs to L (RM), • we have that sss=fen (562, )dy) ds. This is a slightly stronger version of the Fubini's Theorem that we proved in class, for instance one could define fxs and use the...
4. Define the function f: 0,00) +R by the formula f(x) = dt. +1 Comment: The integrand does not have a closed form anti-derivative, so do not try to answer the following questions by computing an anti-derivative. Use some properties that we learned. (a) (4 points). Prove that f(x) > 0 for all x > 0, hence f: (0,00) + (0,0). (b) (4 points). Prove that f is injective. (c) (6 points). Prove that f: (0,00) (0,00) is not surjective,...
3. (a) Let f be an infinitely differentiable function on R and define х F(x) = e-y f(y) dy. Find and prove a formula for F(n), the nth derivative of F. (b) Show that if f is a polynomial then there exists a constant C such that F(n)(x) = Cem for sufficiently large n. Find the least n for which it is true.
10. (10 points) A function f : R2 + R is called a probability density function on D CR if (6) f(, y) 0 for all (x, y) E D and (i) SD. f(x,y)dA= 1. ſk(1 – 22 – y2) 22 + y2 <1 (a) For what constant k is the function f(z,y) a prob- 12 + y2 > 1 ability density function? Note that D= {(1, Y) ER? : x2 + y² <1}, the closed unit disk in R2...
Problem 3. Define the function: 2+_ 0 if (z,y)#10.0) if (a,y)-(0,0) f(x, v)= (a) Graph the top portion of the function using Geogebra. Does the function appear to be continuus at 0? (b) Find fz(z, y) and fy(z, y) when (z, y) #10.0) (c) Find f(0,0) and s,(0,0) using the limit definitions of partial derivatives and f,(0,0)-lim rah) - f(O,0) d) Use these limit definitions to show that fay(0,0)--1, while x(0,0)-1 (e) Can we conclude from Clairaut's theorem that()-yr(x,y) for...
Please answer this question
Implicit Function Theorem in Two Variables: Let g: R2 - R be a smooth function. Set Suppose g(a, b)-0 so that (a, b) є S and dg(a, b) 0. Then there exists an open neighborhood of (a, b) say V such that SnV is the image of a smooth parameterized curve. (1) Verify the implicit function theorem using the two examples above (2) Since dg(a, b)メ0, argue that it suffices to assume a,b)メ0. (3) Prove the...
4. f(x) is a continuous function on [0, 1] and So f (x)dx = a, where a is constant. Evaluate the following double integral f(x)f(y)dydx. (Hint: Change the order of the integration and use the property of the double integral, so that you can apply Fubini's theorem.)
Consider f : [0, 1] x [0, 1] C R2 + R defined by f(x,y) = ſi if y is rational 2x if y is irrational Show that f is not integrable over R by the following steps: in (a) For each n > 1, find a Sn:= Eosi,jan f(a 6? b., in [0, 1] for 0 < i, j < n such that the Riemann sum converges as n + 0.[10 pts] n 1 n2 n i, ja (b)...
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