Problem 2 (5 points) Let f be a continuous function over R, and let g(x) represent...
Problem 1. Consider the function f(x)- 3.12 show that f is Riemann integrable on [0.2] and use the definition to find .后f(x)dr Problem 2. Consider the function -2, zEQ 2, O f(r) = Show that f is not Riemann integrable on 0,1 but s Reemann integrable on this interval. Problem 3. (a) Let f be a real-valued function on a, b] such thatf()0 for all c, where c E [a, b Prove that f is Riemann integrable on a, b...
6. Let f be a continuous function on R and define F(z) = | r-1 f(t)dt x E R. Show that F is differentiable on R and compute F'
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
Problem(2) (5 points) A continuous distribution has density function k sin(r); 0rST 0; f(x) = otherwise. (a) Find the numerical value of k so that f(r) is a density function. (b) Find E[X] (c) Find E[X2. (d) Find Var X] Problem(2) (5 points) A continuous distribution has density function k sin(r); 0rST 0; f(x) = otherwise. (a) Find the numerical value of k so that f(r) is a density function. (b) Find E[X] (c) Find E[X2. (d) Find Var X]
2. Rolle's theorem states that if F : [a, b] → R is a continuous function, differentiable on Ja, bl, and F(a) = F(b) then there exists a cela, b[ such that F"(c) = 0. (a) Suppose g : [a, b] → R is a continuous function, differentiable on ja, bl, with the property that (c) +0 for all cela, b[. Using Rolle's theorem, show that g(a) + g(b). [6 Marks] (b) Now, with g still as in part (a),...
Problem 4. For r E [0, 1, fnd F)-(t)dt, where fr) 3 2r. Verify that F is continuous on [0,1] and F"(z) =f(z) at all points where f is continuous. Problern 5. Suppose that g, h : [c, d] → [a,b] are differentiable. ForエE [c,d] define h(a) Find H'(r) Problem 4. For r E [0, 1, fnd F)-(t)dt, where fr) 3 2r. Verify that F is continuous on [0,1] and F"(z) =f(z) at all points where f is continuous. Problern...
problem 4 plz if r Q (4) Define the function f R -R by f(x) Find all points where f is if E R\Q continuous, and all points where f is differentiable. (No justification needed.)
5. 5 Let f(x) be continuous and differentiable on f(5) = 4. Mark TRUE or FALSE for the following statements and JUSTIFY. (No points will be given without the correct justification) [0, 10] with f(0) = f(10) 0 and (E) There is some c E (0,5) such that f'(c) = 5. 5 Let f(x) be continuous and differentiable on f(5) = 4. Mark TRUE or FALSE for the following statements and JUSTIFY. (No points will be given without the correct...
1) Suppose f (a, b) R is continuous. The Carathéodory Theorem says that f(x) is differentiable at -cE (a, b) if 3 (a, b)-R which is continuous, and so that, (a) Show, for any constant a and continuous function (x), that af(x) is continuous at z-c by finding a Carathéodory function Paf(x). (b) Show, for any constants a, B, that if g : (a, b) -R is differentiable at c, with Carathéodory function pg(z), then the linear combination of functions,...
Exercise 5.3.4: Let f: [a,b] → R be a continuous function. Let ce [a,b] be arbitrary. Define po the Prove that F is differentiable and that F'(x) = f(x) for all x € [a,b]. series on the