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For this problem, you will use the following equation in your calculations, where f(ti) = a,...
Problem 4 (a). 1 Evaluate dx, where i is some positive constant. -5teis +6e-ix Problem 4...
Problem 4 (a). 1 Evaluate dx, where i is some positive constant. -5teis +6e-ix Problem 4 (b). Given that f(x) is continuous everywhere and 7/4 f(2 sect) dt =1010, cost 0 evaluate ) (/*+2+ +5 )dt. --1 Problem 4(c). Let f(x) be continuous on (0,1). T T Show that Į 2 f(sin x)de = / f\sin x)dz. 0 0
4. Consider the boundary value problem defined by the partial differential equation д?и д?и = 0,...
4. Consider the boundary value problem defined by the partial differential equation д?и д?и = 0, ду? y > 0, да? with boundary conditions u(0, y) = u(T,y) = 0, u(x, 0) = 1 and limy-v00 |u(x, y)|< 0o. (a) Use separation of variables to find the eigenvalues and general series solution in terms of the normal modes. (b) Impose the inhomogeneous boundary condition u(x,0) = 1 to find the constants in the general series solution and hence the solution...
step by step please, thank you (2) Use Stokes' Theorem to evaluate the integral F.dr, where...
step by step please, thank
you
(2) Use Stokes' Theorem to evaluate the integral F.dr, where F(x, y, z) =< -Y, I, z > and where S is the upper hemispherical surface defined by z = v1- 2 - y2. The boundary of S is the curve C defined by Cos (t) y= sin (t) 0t 27 Z=0
f 5. (See 2.5, 16) Two models used in population modeling are the Logistic equation and...
f
5. (See 2.5, 16) Two models used in population modeling are the Logistic equation and Gompertz equation: dy dt dy = (r-ay)y and where r,a>0 are constant dt a) For both, sketch the graph of f(y) versus y, find the critical points, and determine asymptotic stability. (b) For 0 y a, determine where the graph of y versus t is concave up and where it is concave down. (c) Sketch solution curves near critical points, discussing differences in the...
Problem 1: (3 +2+3+2 10, sampling) Consider the continuous-time signal x(t) = 3 + cos(10?1+ 5)...
Problem 1: (3 +2+3+2 10, sampling) Consider the continuous-time signal x(t) = 3 + cos(10?1+ 5) + sin(15?), t E R (a) Find the Fourier transform X-Fr. Hint: (F ejuot) (w) 2??(w-wo) (b) What is the Nyquist Frequency wn in radians/s of x? (c) Write an expression for the Fourier transform of the ideal sampling of x with sam- pling period T, = 2n/Cav.), i.e., ?00_ox(AZ)6(t-kZ) Hint: (F eiru>tz(t) (w) - X(w - rus) and recall Poisson's identity, CO eyru'st,...
3. The following utility function is known as CES (constant elasticity of substi- tution) function where...
3. The following utility function is known as CES (constant elasticity of substi- tution) function where α > 0 β > 0 (a) Is this function homothetic? (b) How does the MRS,y depend on the ratio x/y? Specifically, show that the MRSy is strictly decreasing in the ratio x/y for all values δ < 1, increasing in the ratio x/y for all values δ > 1 and constant for δ 1. (c) Show that if x = y, the MRS...
We will all rate if correct 1) For this problem use the following space curve: F(t)...
We will all rate if correct
1) For this problem use the following space curve: F(t) =< t, 3 sin(t), 3 cos(t) > a) Determine the unit tangent vector: T. b) Determine the unit normal vector: Ñ. c) Determine the curvature of this space curve at the point: (0,0,3). d) Determine the arc length of the curve between t = 0 and t = 1.
Prove: By taking the following problem as being given/true : (Analysis on Metric Spaces) Let f...
Prove:
By taking the following problem as being given/true :
(Analysis on Metric Spaces)
Let f : [0, 1] x [0, 1] + R be defined by f(x,y) = ſi if y=x? if y #r? Show that f is integrable on [0, 1] x [0,1]. Let f : [0, 1] + R be uniformly continuous, so that for every e > 0, there exists 8 >0 such that -y<= f(x) - f(y)< € for every I, Y E (0,1). The...
1) For this problem use the following space curve: F(t) =< t, 3 sin(t), 3 cos(t)...
1) For this problem use the following space curve: F(t) =< t, 3 sin(t), 3 cos(t) > a) Determine the unit tangent vector: T. b) Determine the unit normal vector: Ñ. c) Determine the curvature of this space curve at the point: (0,0,3). d) Determine the arc length of the curve between t = 0 and t = 1.
Let f: [0,1]→R be uniformly continuous, so that for every >0, there exists δ >0 such...
Let f: [0,1]→R be uniformly continuous, so that for every >0,
there exists δ >0 such that |x−y|< δ=⇒|f(x)−f(y)|< for
every x,y∈[0,1].The graph of f is the set G f={(x,f(x))
:x∈[0,1]}.Show that G f has measure
zero
Let f : [0, 1] → R be uniformly continuous, so that for every e > 0, there exists 8 >0 such that 2- y<83|f() - f(y)< € for every 1, 9 € [0,1]. The graph of f is the set Gj =...
Problem 4 (a). 1 Evaluate dx, where i is some positive constant. -5teis +6e-ix Problem 4 (b). Given that f(x) is continuous everywhere and 7/4 f(2 sect) dt =1010, cost 0 evaluate ) (/*+2+ +5 )dt. --1 Problem 4(c). Let f(x) be continuous on (0,1). T T Show that Į 2 f(sin x)de = / f\sin x)dz. 0 0
4. Consider the boundary value problem defined by the partial differential equation д?и д?и = 0, ду? y > 0, да? with boundary conditions u(0, y) = u(T,y) = 0, u(x, 0) = 1 and limy-v00 |u(x, y)|< 0o. (a) Use separation of variables to find the eigenvalues and general series solution in terms of the normal modes. (b) Impose the inhomogeneous boundary condition u(x,0) = 1 to find the constants in the general series solution and hence the solution...
step by step please, thank
you
(2) Use Stokes' Theorem to evaluate the integral F.dr, where F(x, y, z) =< -Y, I, z > and where S is the upper hemispherical surface defined by z = v1- 2 - y2. The boundary of S is the curve C defined by Cos (t) y= sin (t) 0t 27 Z=0
f
5. (See 2.5, 16) Two models used in population modeling are the Logistic equation and Gompertz equation: dy dt dy = (r-ay)y and where r,a>0 are constant dt a) For both, sketch the graph of f(y) versus y, find the critical points, and determine asymptotic stability. (b) For 0 y a, determine where the graph of y versus t is concave up and where it is concave down. (c) Sketch solution curves near critical points, discussing differences in the...
Problem 1: (3 +2+3+2 10, sampling) Consider the continuous-time signal x(t) = 3 + cos(10?1+ 5) + sin(15?), t E R (a) Find the Fourier transform X-Fr. Hint: (F ejuot) (w) 2??(w-wo) (b) What is the Nyquist Frequency wn in radians/s of x? (c) Write an expression for the Fourier transform of the ideal sampling of x with sam- pling period T, = 2n/Cav.), i.e., ?00_ox(AZ)6(t-kZ) Hint: (F eiru>tz(t) (w) - X(w - rus) and recall Poisson's identity, CO eyru'st,...
3. The following utility function is known as CES (constant elasticity of substi- tution) function where α > 0 β > 0 (a) Is this function homothetic? (b) How does the MRS,y depend on the ratio x/y? Specifically, show that the MRSy is strictly decreasing in the ratio x/y for all values δ < 1, increasing in the ratio x/y for all values δ > 1 and constant for δ 1. (c) Show that if x = y, the MRS...
We will all rate if correct
1) For this problem use the following space curve: F(t) =< t, 3 sin(t), 3 cos(t) > a) Determine the unit tangent vector: T. b) Determine the unit normal vector: Ñ. c) Determine the curvature of this space curve at the point: (0,0,3). d) Determine the arc length of the curve between t = 0 and t = 1.
Prove:
By taking the following problem as being given/true :
(Analysis on Metric Spaces)
Let f : [0, 1] x [0, 1] + R be defined by f(x,y) = ſi if y=x? if y #r? Show that f is integrable on [0, 1] x [0,1]. Let f : [0, 1] + R be uniformly continuous, so that for every e > 0, there exists 8 >0 such that -y<= f(x) - f(y)< € for every I, Y E (0,1). The...
1) For this problem use the following space curve: F(t) =< t, 3 sin(t), 3 cos(t) > a) Determine the unit tangent vector: T. b) Determine the unit normal vector: Ñ. c) Determine the curvature of this space curve at the point: (0,0,3). d) Determine the arc length of the curve between t = 0 and t = 1.
Let f: [0,1]→R be uniformly continuous, so that for every >0,
there exists δ >0 such that |x−y|< δ=⇒|f(x)−f(y)|< for
every x,y∈[0,1].The graph of f is the set G f={(x,f(x))
:x∈[0,1]}.Show that G f has measure
zero
Let f : [0, 1] → R be uniformly continuous, so that for every e > 0, there exists 8 >0 such that 2- y<83|f() - f(y)< € for every 1, 9 € [0,1]. The graph of f is the set Gj =...
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