THIS QUESTION MUST BE ANSWER
USING TAYLOR SERIES

THIS QUESTION MUST BE ANSWER USING TAYLOR SERIES 2. (10 pts) Let I be an open...
Let U be an open subset of R". Let f: UCR" ->Rm. (a) Prove that f is continuously differentiable if and only if for each a e U, for eache > 0, there exists o > 0 such that for each xe U, if ||x - a| << ô, then |Df (x) Df(a)| < e.
please i need the question 8.(a)(b) for the detailed proof and
explaination ! thanks !
Let B fE C (R, R) | f(x)> 0 for all E R (a) Is B open? If not, what is B°? (b) What is B?
Suppose that UCC is open and connected and a E U. Let F:= {f € H(U)| Re(f)> 0, f(a) 1} Show that Fis normal.
Exercise 7. Let X be a standard normal random variable. Prove that for any integer n > 0, ELY?"] = 1207) and E[x2n+] = 0.
Evaluate the following limit using Taylor series. 3 lim 2x2 zle x2 1 X>00
Prove or Disprove:
Let p E P(F) and suppose that deg p > 1 and p is irreducible. Then p(a)メ0 for all a E F.
Let z=5 where x, y, z E R. Prove that z? +z2+z?>
For all n E N prove that 0 <e- > < 2 k!“ (n + 1)! k=0 Hint: Think about Taylor approximations of the function e".
(2) Prove that the following are equivalent for x ER and A CR. (a) X E A. Here A denotes the closure of A. (b) For every e > 0, N(x; e) n A +0. (c) For every open set U, if r EU then UNA+.
Use Definition 7.1.1. DEFINITION 7.1.1 Laplace Transform Let f be a function defined for t > 0. Then the integral Kf(t)} = [e-stf(t) dt is said to be the Laplace transform of f, provided that the integral converges. Find L{f(t)}. (Write your answer as a function of s.) f(t) = {6. Ost<3 PROI} = (s > 0)