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plz help me !! Thanks 1. In class we showed that the function f : R → R given by (if>o 0 if a S0 was smooth (but not...
what I need for is #2! #1 is attached for #2. Please help me! Thanks 1. In class we showed that the function f : R → R...
what I need for is #2!
#1 is attached for #2.
Please help me! Thanks
1. In class we showed that the function f : R → R given by (if>o 0 if a S0 was smooth (but not real analytic). Note that f(x) approaches a horizontal asymptote (y = 1) as a goes to positive infinity. (a) Show that f(x)+f(1-2)メ0 for all x E R, so that g : R → R given by g(x)- 70 is also a...
plz help me analysis question! Thanks in advance 2. Let h : R-+ R be the smooth function given by h(z) g is as in Probl...
plz help me analysis question!
Thanks in advance
2. Let h : R-+ R be the smooth function given by h(z) g is as in Problem 1 g(z + 2g(2-x) for all r E R, where (a) Show that if a < -2 0 g(2) if -2< <-1 h(x) if 2 0 (b) Use part (d) of Proble 1 to show that for all E 0,9 in fact for all ,. Conclude that for all e 0,1 The functions from...
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...
Example 8.5.1. Let if 0< x< T if 0 or r? -1 if -т <т < 0. 1 f(x)= 0 _ The fact that f is an odd function (i...
Example 8.5.1. Let if 0< x< T if 0 or r? -1 if -т <т < 0. 1 f(x)= 0 _ The fact that f is an odd function (i.e., f(-x) = -f(x)) means we can avoid doing any integrals for the moment and just appeal to a symmetry argument to conclude T f (x) cos(nar)dx 0 and an f(x)dax = 0 ao -- T 27T -T for all n 1. We can also simplify the integral for bn by...
Orthogonal projections. In class we showed that if V is a finite-dimensional inner product space ...
Orthogonal projections. In class we showed that if V is a finite-dimensional inner product space and U-V s a subspace, then U㊥ U↓-V, (U 1-U, and Pb is well-defined Inspecting the proofs, convince yourself that all that was needed was for U to be finite- dimensional. (In fact, your book does it this way). Then answer the following questions (a) Let V be an inner product space. Prove that for any u V. if u 0, we have proj, Pspan(v)...
-100x 1. Given the function f(x)=- (1-0.5x) (a) Find the y-intercept point (if there is any):...
-100x 1. Given the function f(x)=- (1-0.5x) (a) Find the y-intercept point (if there is any): (b) Find the x-intercept point(s) (if there is any): (c) Find f'(x): (d) Find critical number(s) of f(x) (Type 1 and Type 2, if there are any): (e) Find the critical point(s) (if there are any): (f) Find the open x-intervals where f(x) increases and decreases: (g) Find the behavior of the function for very large positive x-values (find limit as x goes to...
3. Consider the function defined by f(x) = 1, 0 < r< a, | 0, a<...
3. Consider the function defined by f(x) = 1, 0 < r< a, | 0, a< x < T, where 0a < T (a) Sketch the odd and even periodic extension of f (x) on the interval -3n < x < 3« for aT/2 (b) Find the half-range Fourier sine series expansion of f(x) for arbitrary a. (e) To what value does the half-range Fourier sine series expansion converge at r a? [8 marks
3. Consider the function defined by...
1. Consider the function defined by 1- x2, 0< |x| < 1, f(x) 0, and f(r) f(x+4) (a) Sketch the graph of f(x) on th...
1. Consider the function defined by 1- x2, 0< |x| < 1, f(x) 0, and f(r) f(x+4) (a) Sketch the graph of f(x) on the interval -6, 6] (b) Find the Fourier series representation of f(x). You must show how to evaluate any integrals that are needed 2. Consider the function 0 T/2, T/2, T/2 < T. f(x)= (a) Sketch the odd and even periodic extension of f(x) for -3r < x < 3m. (b) Find the Fourier cosine series...
plz help me analysis question! Thanks in advance 5. For each n є N let fn : R R be given by f,(x)-imrz. Prove that the...
plz help me analysis question! Thanks in advance
5. For each n є N let fn : R R be given by f,(x)-imrz. Prove that the sequence {f. of functions converges pointwise to the function f R- R given by 1+nr if x#0 f(x)-0
5. For each n є N let fn : R R be given by f,(x)-imrz. Prove that the sequence {f. of functions converges pointwise to the function f R- R given by 1+nr if x#0 f(x)-0
what I need for is #2!
#1 is attached for #2.
Please help me! Thanks
1. In class we showed that the function f : R → R given by (if>o 0 if a S0 was smooth (but not real analytic). Note that f(x) approaches a horizontal asymptote (y = 1) as a goes to positive infinity. (a) Show that f(x)+f(1-2)メ0 for all x E R, so that g : R → R given by g(x)- 70 is also a...
plz help me analysis question!
Thanks in advance
2. Let h : R-+ R be the smooth function given by h(z) g is as in Problem 1 g(z + 2g(2-x) for all r E R, where (a) Show that if a < -2 0 g(2) if -2< <-1 h(x) if 2 0 (b) Use part (d) of Proble 1 to show that for all E 0,9 in fact for all ,. Conclude that for all e 0,1 The functions from...
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...
Example 8.5.1. Let if 0< x< T if 0 or r? -1 if -т <т < 0. 1 f(x)= 0 _ The fact that f is an odd function (i.e., f(-x) = -f(x)) means we can avoid doing any integrals for the moment and just appeal to a symmetry argument to conclude T f (x) cos(nar)dx 0 and an f(x)dax = 0 ao -- T 27T -T for all n 1. We can also simplify the integral for bn by...
Orthogonal projections. In class we showed that if V is a finite-dimensional inner product space and U-V s a subspace, then U㊥ U↓-V, (U 1-U, and Pb is well-defined Inspecting the proofs, convince yourself that all that was needed was for U to be finite- dimensional. (In fact, your book does it this way). Then answer the following questions (a) Let V be an inner product space. Prove that for any u V. if u 0, we have proj, Pspan(v)...
-100x 1. Given the function f(x)=- (1-0.5x) (a) Find the y-intercept point (if there is any): (b) Find the x-intercept point(s) (if there is any): (c) Find f'(x): (d) Find critical number(s) of f(x) (Type 1 and Type 2, if there are any): (e) Find the critical point(s) (if there are any): (f) Find the open x-intervals where f(x) increases and decreases: (g) Find the behavior of the function for very large positive x-values (find limit as x goes to...
3. Consider the function defined by f(x) = 1, 0 < r< a, | 0, a< x < T, where 0a < T (a) Sketch the odd and even periodic extension of f (x) on the interval -3n < x < 3« for aT/2 (b) Find the half-range Fourier sine series expansion of f(x) for arbitrary a. (e) To what value does the half-range Fourier sine series expansion converge at r a? [8 marks
3. Consider the function defined by...
1. Consider the function defined by 1- x2, 0< |x| < 1, f(x) 0, and f(r) f(x+4) (a) Sketch the graph of f(x) on the interval -6, 6] (b) Find the Fourier series representation of f(x). You must show how to evaluate any integrals that are needed 2. Consider the function 0 T/2, T/2, T/2 < T. f(x)= (a) Sketch the odd and even periodic extension of f(x) for -3r < x < 3m. (b) Find the Fourier cosine series...
plz help me analysis question! Thanks in advance
5. For each n є N let fn : R R be given by f,(x)-imrz. Prove that the sequence {f. of functions converges pointwise to the function f R- R given by 1+nr if x#0 f(x)-0
5. For each n є N let fn : R R be given by f,(x)-imrz. Prove that the sequence {f. of functions converges pointwise to the function f R- R given by 1+nr if x#0 f(x)-0
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