Homework Answers
2 er Let I be an interval of R, and define the function f :I→ R by f(x) 1 +e2z or every z EZ. (a) Find the largest...
2 er Let I be an interval of R, and define the function f :I→ R by f(x) 1 +e2z or every z EZ. (a) Find the largest interval T where f is strictly increasing. (b) For this interval Z, determine the range f(T) (c) Let T- f(I). Show that the function f : I -» T is injective and surjective. (d) Determine the inverse function f-i : T → 1. (e) Verify that (fo f-1)()-y for every y E...
5. Let f(x) = arctan(In x) for all x >0. A graph of y = f(x)...
5. Let f(x) = arctan(In x) for all x >0. A graph of y = f(x) is shown in the figure. (a) Find the formula for the derivative f'(x). Then explain how you can deduce from this formula that f is invertible. (b) Find the formula for f-1(x), the inverse of f. (c) What is the domain and range of f-1? (d) Sketch a graph of the function y=f-1(x). (e) Now determine the value of (F-1)(0) using your results from...
C2.1 (Probability integral transform.) Let X be a random variable with cu mulative distribution function F,...
C2.1 (Probability integral transform.) Let X be a random variable with cu mulative distribution function F, and suppose that F is continuous and strictly increasing on R. (i) Show that F has a well-defined inverse function G : (0,1) → R, which is (ii) Using G, or otherwise, show that the random variable F(X) is uniformly 시 strictly increasing distributed on [0,1
Let R be an interval (open, closed, neither are all fine) and let f: I-> R be a continuous strictly increasing funct...
Let R be an interval (open, closed, neither are all fine) and let f: I-> R be a continuous strictly increasing function. Do the following: (1) Show that the inverse function f -1 exists. (2) Prove that f is an open map (in the relative topology on I) (3) Prove that f1 is continuous
Let R be an interval (open, closed, neither are all fine) and let f: I-> R be a continuous strictly increasing function. Do the following: (1)...
3. Let the function f be a real valued bounded continuous function on R. Prove that...
3. Let the function f be a real valued bounded continuous function on R. Prove that there is a solution of the equation f(x) = x, xER. Now choose a number a with f(a) > a and define the sequence (an) recursively by defining al = a and a叶1 = f(an), where n E N. If f is strictly increasing on R, show that (an) converges to a solution of the equation (0.1). This method for approximating the solution is...
Exercise 31: (Chain rule) Let g : la,b] → R be differentiable and strictly increasing and f : R-IR be continuous. Show...
Exercise 31: (Chain rule) Let g : la,b] → R be differentiable and strictly increasing and f : R-IR be continuous. Show that gr) F(x) :=| f(t)dt Jg(a) is differentiable and compute its derivative
Exercise 31: (Chain rule) Let g : la,b] → R be differentiable and strictly increasing and f : R-IR be continuous. Show that gr) F(x) :=| f(t)dt Jg(a) is differentiable and compute its derivative
1. Let f: R\{-1} → R, f() = 2+1 (a) Prove that f is not an...
1. Let f: R\{-1} → R, f() = 2+1 (a) Prove that f is not an increasing function on its domain, but its restrictions to intervals fl-20.-1) and fl(-1,00) are strictly increasing. (b) Find a codomain for fl-1.00) that makes the function bijective. Find the composi- tional inverse of our function. Sketch both our function and its inverse on the same set of axes.
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...
#55, 59 In Exercises 55 and 56, determine whether the statement is true or false. If...
#55, 59
In Exercises 55 and 56, determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. 55. If f is continuous at x = a, then f is differentiable at x = a. 56. If f is continuous at x = a and g is differentiable at x = a, then lim f(x)g(x) = f(a)g(a). X 57. Sketch the...
2. Let X be a continuous r.v. with pdf f () and cdf F(x). Let U...
2. Let X be a continuous r.v. with pdf f () and cdf F(x). Let U F (X). Show that, as long as F(x) is strictly monotonic increasing, U is uniformly distributed on (0,1). Discuss why this result is important, given that it is known how to simulate Uniformly distributed random variables easily.
2 er Let I be an interval of R, and define the function f :I→ R by f(x) 1 +e2z or every z EZ. (a) Find the largest interval T where f is strictly increasing. (b) For this interval Z, determine the range f(T) (c) Let T- f(I). Show that the function f : I -» T is injective and surjective. (d) Determine the inverse function f-i : T → 1. (e) Verify that (fo f-1)()-y for every y E...
5. Let f(x) = arctan(In x) for all x >0. A graph of y = f(x) is shown in the figure. (a) Find the formula for the derivative f'(x). Then explain how you can deduce from this formula that f is invertible. (b) Find the formula for f-1(x), the inverse of f. (c) What is the domain and range of f-1? (d) Sketch a graph of the function y=f-1(x). (e) Now determine the value of (F-1)(0) using your results from...
C2.1 (Probability integral transform.) Let X be a random variable with cu mulative distribution function F, and suppose that F is continuous and strictly increasing on R. (i) Show that F has a well-defined inverse function G : (0,1) → R, which is (ii) Using G, or otherwise, show that the random variable F(X) is uniformly 시 strictly increasing distributed on [0,1
Let R be an interval (open, closed, neither are all fine) and let f: I-> R be a continuous strictly increasing function. Do the following: (1) Show that the inverse function f -1 exists. (2) Prove that f is an open map (in the relative topology on I) (3) Prove that f1 is continuous
Let R be an interval (open, closed, neither are all fine) and let f: I-> R be a continuous strictly increasing function. Do the following: (1)...
3. Let the function f be a real valued bounded continuous function on R. Prove that there is a solution of the equation f(x) = x, xER. Now choose a number a with f(a) > a and define the sequence (an) recursively by defining al = a and a叶1 = f(an), where n E N. If f is strictly increasing on R, show that (an) converges to a solution of the equation (0.1). This method for approximating the solution is...
Exercise 31: (Chain rule) Let g : la,b] → R be differentiable and strictly increasing and f : R-IR be continuous. Show that gr) F(x) :=| f(t)dt Jg(a) is differentiable and compute its derivative
Exercise 31: (Chain rule) Let g : la,b] → R be differentiable and strictly increasing and f : R-IR be continuous. Show that gr) F(x) :=| f(t)dt Jg(a) is differentiable and compute its derivative
1. Let f: R\{-1} → R, f() = 2+1 (a) Prove that f is not an increasing function on its domain, but its restrictions to intervals fl-20.-1) and fl(-1,00) are strictly increasing. (b) Find a codomain for fl-1.00) that makes the function bijective. Find the composi- tional inverse of our function. Sketch both our function and its inverse on the same set of axes.
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...
#55, 59
In Exercises 55 and 56, determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. 55. If f is continuous at x = a, then f is differentiable at x = a. 56. If f is continuous at x = a and g is differentiable at x = a, then lim f(x)g(x) = f(a)g(a). X 57. Sketch the...
2. Let X be a continuous r.v. with pdf f () and cdf F(x). Let U F (X). Show that, as long as F(x) is strictly monotonic increasing, U is uniformly distributed on (0,1). Discuss why this result is important, given that it is known how to simulate Uniformly distributed random variables easily.
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