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3) (Assessment Topic: Continuity) Consider the function sgn(x)- (a) What is the domain of sgn(x)?...
A1. A few useful properties of the Dirac delta function The Dirac δ(z) functions is defined...
A1. A few useful properties of the Dirac delta function The Dirac δ(z) functions is defined by δ(z) = 0 if |メ0, δ(z) = oo ifx-0 but the integral of the function over any interval containing the zero of the argument is unity, Equivalent, if f(x) is continuous at the origin One should treat the Dirac δ function as a limit of a sequence of functions peaked at x-0 As the limit is approached, the height of the peak increases...
1. Define the function sgn by: ifx>0 ifx=0 sgn(x) = 0 Now define h(x): [0,1]R by...
1. Define the function sgn by: ifx>0 ifx=0 sgn(x) = 0 Now define h(x): [0,1]R by 51 if0cz ifx=0 h(z) =(sgn(sin(1/4)) i Prove that h(x) is integrable.
real analysis 1,3,8,11,12 please 4.4.3 4.4.11a Limits and Continuity 4 Chapter Remark: In the statement of T...
real analysis
1,3,8,11,12 please
4.4.3
4.4.11a
Limits and Continuity 4 Chapter Remark: In the statement of Theorem 4.4.12 we assumed that f was tone and continuous on the interval I. The fact that f is either stric tric. strictly decreasing on / implies that f is one-to-one on t one-to-one and continuous on an interval 1, then as a consequence of the value theorem the function f is strictly monotone on I (Exercise 15). This false if either f is...
2. Prove the following useful properties of Dirac δ-functions (a) δ(ax) = (z) (b) zfic) =0 (c) f(x)5(-a) f(a)5(r d) δ(z-a (aメ0) a) ( dz9(x-a) ) where θ(x) is the step function defined as 1 if r 0 0...
2. Prove the following useful properties of Dirac δ-functions (a) δ(ax) = (z) (b) zfic) =0 (c) f(x)5(-a) f(a)5(r d) δ(z-a (aメ0) a) ( dz9(x-a) ) where θ(x) is the step function defined as 1 if r 0 0 if r <0 θ(z) =
2. Prove the following useful properties of Dirac δ-functions (a) δ(ax) = (z) (b) zfic) =0 (c) f(x)5(-a) f(a)5(r d) δ(z-a (aメ0) a) ( dz9(x-a) ) where θ(x) is the step function defined as 1 if...
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)...
Question 2: (20 points) Consider the function signum Find the general global solution of the differential equation y" + (sgn x)y - 0. N.B. The general global solution is a function y: RR that is...
Question 2: (20 points) Consider the function signum Find the general global solution of the differential equation y" + (sgn x)y - 0. N.B. The general global solution is a function y: RR that is twice differentiable and verifies the differential equation (1) on R.
Question 2: (20 points) Consider the function signum Find the general global solution of the differential equation y" + (sgn x)y - 0. N.B. The general global solution is a function y: RR that is...
Consider the function y=f(x) whose graph is given below. Identify the following: A. domain: B. range:...
Consider the function y=f(x) whose graph is given below. Identify the following: A. domain: B. range: c. lim f(2)= D. lim $(=) E lim f(x) & lim f(x) G. lim f(z)- H. lim f(z) 1. lim f(1) J. Lim f(x)= K vertical asymptote(s): L. horizontal asymptote(s):
7 points Question 3. An Unusual Integrable Function (Show Working) Consider the function f : 10, ...
7 points Question 3. An Unusual Integrable Function (Show Working) Consider the function f : 10, 11 → R defined by 1 if r-for some nEN; f(x) = 0 for all other x E [0,1 (1 subpts) (a) Draw a rough diagram of the graph of f. When we study the formal definition of the continuity of a function later in the course, we will be able to prove that this function is discontinuous at those domain values r such...
Let f(x) be a continous function defined on R. Consider the following function, g(x) = max{f(t)\t...
Let f(x) be a continous function defined on R. Consider the following function, g(x) = max{f(t)\t € [2 – 1, 2+1]}. Prove that g(x) is also continous. Hint: To prove g(x) is continous at x = xo. You can consider the continuity of f(x) at the two boundary point xo - 1 and xo +1. When x get close to xo, the points in (7 - 1, + 1) is close to xo - 1, xo + 1, or inside...
A1. A few useful properties of the Dirac delta function The Dirac δ(z) functions is defined by δ(z) = 0 if |メ0, δ(z) = oo ifx-0 but the integral of the function over any interval containing the zero of the argument is unity, Equivalent, if f(x) is continuous at the origin One should treat the Dirac δ function as a limit of a sequence of functions peaked at x-0 As the limit is approached, the height of the peak increases...
1. Define the function sgn by: ifx>0 ifx=0 sgn(x) = 0 Now define h(x): [0,1]R by 51 if0cz ifx=0 h(z) =(sgn(sin(1/4)) i Prove that h(x) is integrable.
real analysis
1,3,8,11,12 please
4.4.3
4.4.11a
Limits and Continuity 4 Chapter Remark: In the statement of Theorem 4.4.12 we assumed that f was tone and continuous on the interval I. The fact that f is either stric tric. strictly decreasing on / implies that f is one-to-one on t one-to-one and continuous on an interval 1, then as a consequence of the value theorem the function f is strictly monotone on I (Exercise 15). This false if either f is...
2. Prove the following useful properties of Dirac δ-functions (a) δ(ax) = (z) (b) zfic) =0 (c) f(x)5(-a) f(a)5(r d) δ(z-a (aメ0) a) ( dz9(x-a) ) where θ(x) is the step function defined as 1 if r 0 0 if r <0 θ(z) =
2. Prove the following useful properties of Dirac δ-functions (a) δ(ax) = (z) (b) zfic) =0 (c) f(x)5(-a) f(a)5(r d) δ(z-a (aメ0) a) ( dz9(x-a) ) where θ(x) is the step function defined as 1 if...
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)...
Question 2: (20 points) Consider the function signum Find the general global solution of the differential equation y" + (sgn x)y - 0. N.B. The general global solution is a function y: RR that is twice differentiable and verifies the differential equation (1) on R.
Question 2: (20 points) Consider the function signum Find the general global solution of the differential equation y" + (sgn x)y - 0. N.B. The general global solution is a function y: RR that is...
Consider the function y=f(x) whose graph is given below. Identify the following: A. domain: B. range: c. lim f(2)= D. lim $(=) E lim f(x) & lim f(x) G. lim f(z)- H. lim f(z) 1. lim f(1) J. Lim f(x)= K vertical asymptote(s): L. horizontal asymptote(s):
7 points Question 3. An Unusual Integrable Function (Show Working) Consider the function f : 10, 11 → R defined by 1 if r-for some nEN; f(x) = 0 for all other x E [0,1 (1 subpts) (a) Draw a rough diagram of the graph of f. When we study the formal definition of the continuity of a function later in the course, we will be able to prove that this function is discontinuous at those domain values r such...
Let f(x) be a continous function defined on R. Consider the following function, g(x) = max{f(t)\t € [2 – 1, 2+1]}. Prove that g(x) is also continous. Hint: To prove g(x) is continous at x = xo. You can consider the continuity of f(x) at the two boundary point xo - 1 and xo +1. When x get close to xo, the points in (7 - 1, + 1) is close to xo - 1, xo + 1, or inside...
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