Suppose that is a bounded function with following Lower and Upper Integrals:
and
a) Prove that for every , there exists a partition of such that the difference between the upper and lower sums satisfies .
b) Furthermore, does there have to be a subdivision such that . Either prove it or find a counterexample and show to the contrary.
Suppose that is a bounded function with following Lower and Upper Integrals: and a) Prove...
sin 0, cos 0 Name the quadrant in which the angle lies We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image
Prove, or give a counter example to disprove the following statements. a) b) We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image
Let f(x)= if , if if a) What is the fomain of f(x)? Write in interval notation. b) Determine the y-intercept of the function, if any. Make sure to justify your answer. c) Determine the x-intercepts of the function, if any. Justify your answer. We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this...
Prove that for every positive real (important: is not necessarily an integer), that . Hint: For every , the function is strictly growing. We were unable to transcribe this imageWe were unable to transcribe this imagebe(n") (n log, n) > 0 n
Prove the ratio test . What does this tell you if exists? (Ratio test) If for all sufficiently large n and some r < 1, then converges absolutely; while if for all sufficiently large n, then diverges. lim |.1n+1/01 700 In+1/xn < We were unable to transcribe this image2x+1/2 > 1 We were unable to transcribe this image
Prove the following Let with Then: i) if and only if where the double inequality means and ii) If , if and only if . -2, E ER We were unable to transcribe this imageWe were unable to transcribe this image-E <<E, We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imagea ER We were unable to transcribe this imageWe were unable to transcribe this image
Let X be a banach space such that X= C([a,b]) where - ab+ with the sup norm. Let x and f X. Show that the non linear integral equation u(x) = (sin u(y) dy + f(x) ) has a solution u X. (the integral is from a to b). We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe...
Suppose is a bounded function for which there exists a partition such that . Prove: is a constant function f : a, b] →R We were unable to transcribe this imageL(P, f,a) = U(P, f,a) We were unable to transcribe this image
Write A function Markov that take ?, ? and ? as inputs and return the upper bounds for ?(?≥?⋅??) given by the below Markov inequalities as output. For the binomial distribution with mean and variance , we would like to upper bound the probability for . Example: Markov(100.,0.2,1.5) Output: 0.6666666666666666 Which of the following is the correct output for Markov(200.,0.3,1.1)? A. 0.909 B 0.558 C. 0.986 D. 0872 p.n We were unable to transcribe this imageWe were unable to transcribe...
Show that a bounded and monotone sequence converges. Here a sequence is called monotone, if it is either monotone increasing, that is for all or monotone decreasing, in which case for all . in Sn=1 An+1 > an neN an+1 < an We were unable to transcribe this image