1. [4 pts) Formally prove (i.e. using € - 8 definition) that the function x3 is...
PLEASE ANSWER ALL! SHOWS STEPS 2. (a) Prove by using the definition of convergence only, without using limit theo- (b) Prove by using the definition of continuity, or by using the є_ó property, that 3. Let f be a twice differentiable function defined on the closed interval [0, 1]. Suppose rems, that if (S) is a sequence converging to s, then lim, 10 2 f (x) is a continuous function on R r,s,t e [0,1] are defined so that r...
+1 4. Consider the function ISO 0<<1 -1 = 1 0 1<*52 (x - 2)2 => 2 (a) (10) Use the definition of the limit of a function at a point to evaluate with proof (b) (10) Use the definition of continuity at a point to prove that /(x) is not continuous at -1. (e) (2) Is /(x) uniformly continuous on (-1,2)? If it is, prove it. Other- wise, explain why not. (d) (8) Is f() uniformly continuous on (1,3)?...
1. [15 pts] Use Definition 1.5 (definition of probability function) to prove Propo- sition 1.3 () 15 pts) & (iv) [10 pts). You do not need to prove (i) and (ii). [Definition 1.5/ Let Ω be a set of all possible events. A probability function P : Ω → 0,11 satisfies the follouing three conditions (i) 0s P(A) S 1 for any event A; (iii) For any sequence of mutually exclusive events A1, A2 ,A", i.e. A, n Aj =...
Formally prove the following four statements (i.e., show a constant c and a no such that ): I. 2n is Θ(2n+1) 2. 3 is O(1) 3. 3n2 +4-2n is O(n3) 4. Σί-01 is Ω(n)
[4 Pts. Use the definition of continuity to show that the function f is continuous at <=0 10 g(x)= 3-4
Question 1 1. [5 pts] Give a complete definition of lim f(x) = -oo if... 2. [25 pts] Give an example of each of the following, or state one or more theorems which show that such an example is impossible: a. A countable collection of nonempty closed proper subsets of R whose union is open. b. A nonempty bounded subset of R with no cluster points. c. A convergent sequence with two convergent subsequences with distinct limits. d. A function...
Let f:D + R be a function. (a) Recall the definition that f is uniformly continuous on D. (You do not need to write this down. This only serves as a hint for next parts.) (b) Use (a) and the mean value theorem to prove f(x) = e-% + sin x is uniformly continuous on (0, +00). (c) Use the negation of (a) to prove f(x) = x2 is not uniformly continuous on (0,0).
9. Prove that the function f(x) = ax+b is uniformly continuous on R by directly applying the e, 8 definition of uniform continuity.
the function y=f(x)={ 0-4), 14x+16, x20 x<0 Consider 1. (a) Sketch the graph off. (3 pts.) (b) Verify that the function is continuous everywhere using the properties of the definition and possibly calculating the limit at a particular point. (2 pts.) (c) Show f'(x) is not continuous at x-0. (5 pts.) the function y=f(x)={ 0-4), 14x+16, x20 x
ad cal 2 . Problem 8, (10 pts.) Prove that on the interval [0,0.8) -2n lim dx Problem 9, (10 pts.) na(1-z)". Let fn (z) Prove that . Problem 10, (10 pts.) Using Method of mathematical induction prove that: If function u(x) is such that a,--u then a ,u u, 2n1 . Problem 8, (10 pts.) Prove that on the interval [0,0.8) -2n lim dx Problem 9, (10 pts.) na(1-z)". Let fn (z) Prove that . Problem 10, (10 pts.)...