Let fa(x)= x^a if x>0, 0 if x<0.
a) for what values of a is f continuous at 0?
WHY is a<0 not continuous at 0? Please explain.
Let fa(x)= x^a if x>0, 0 if x<0. a) for what values of a is f continuous at 0? WHY is a<...
plz explain why [ºs(dr diverges, then the integral ["f(z)dz also d) If f(x) is continuous for all x and diverges for any a > 0. Answer: True False 0.8P(1 -0.001P). Then lim P(t) = 800. dP (e) Consider the logistic model dt Answer: True False (f) Let fa. denote the average value of over the interval (a, b). Then - Blac) + Sic for all ce(a,b). Answer: True False
(15 points) Let X be a continuous random variable with cumulative distribution function F(x) = 0, r <α Inr, a< x <b 1, b< (a) Find the values of a and b so that F(x) is the distribution function of a continuous random variable. (b) Find P(X > 2). (c) Find the probability density function f(x) for X. (d) Find E(X)
Let f(x) = cxe-x if x 20 and f(x) = 0 if x < 0. (a) For what value of c is fa probability density function? (b) For that value of c, find P(1<x< 4). 0.368
For what values of a is f(x) continuous for all values x? f(x) = { asinx x+a for χ π for x>π
Let f(x) = x^(1/3) with domain (0,infinity). Prove, by epsilon-delta language, that f is continuous at c in an element of (0, infinity). 2. Let f(0) = 25 with domain (0,00). Prove, by the e-8 language, that f is continuous at CE (0,0)
7, Let X be a continuous random variable with probability density function: 0, f x<0 150 f x> 10 ind ihe avnanted value and mode of random variable X
Suppose f is a continuous and differentiable function on [0,1] and f(0)= f(1). Let a E (0, 1). Suppose Vr,y(0,1) IF f'(x) 0 and f'(y) ±0 THEN f'(x) af'(y) Show that there is exactly f(ax) and f'(x) 0 such that f(x) one Hint: Suppose f(x) is a continuous function on [0, 1] and f(0) x € (0, 1) such that f(x) = f(ax) f(1). Let a e (0,1), there exists an Suppose f is a continuous and differentiable function on...
Let A ⊆ Xτ and let f : Xτ → Yν be topologically continuous. If x is a limit point of A, must f(x) be a limit point of f(A) ⊆ Y ? Explain.
(0, 1) given by f (x) - sin (). Is f Let f b e the function t on the domain uniformly continuous? Explain. (You may take it as given that sin is a continuous function) Suppose that f [0, oo) -R is a continuous function, and suppose also that lim, ->oo f (x)- 0. Prove that f is uniformly continuous Just to be clear: to say that lim,->o f (x) - 0 means that
Answer C 6. Let f be a continuous function on [0, oo) such that 0 f(z) Cl- for some C,e> 0, and let a = fo° f(x) da. (The estimate on f implies the convergence of this integral.) Let fk(x) = kf(ka) a. Show that lim00 fk(x) = 0 for all r > 0 and that the convergence is uniform on [8, oo) for any 6> 0. b. Show that limk00 So ()dz = a. c. Show that lim00 So...