4. (a) Observe that (-oo, b] (-00, ajU (a, b] with (-o0, aln(a, b 0. This...
(12) Suppose that f [0, oo) - [0, o0) and that f E R(0, n), for every n E N. Prove that f is Lebesgue measurable, the Lebesgue integral So.o)dexists, and f dA f (x)dax lim -- noo 0,00)
0, oo) which converges to a certain real Let f be a real-valued continuous function over o0, i.e., lim f(x) = A. Answer the following questions value A as Find the following limit lim aoo a2 f (x)dx 0, oo) which converges to a certain real Let f be a real-valued continuous function over o0, i.e., lim f(x) = A. Answer the following questions value A as Find the following limit lim aoo a2 f (x)dx
(12) Suppose that f: [0, o0) - (0, 00) and that f e R((0, n]), for every n E N. Prove that f is Lebesgue measurable, the Lebesgue integral Jo.0)f dA exists, and f dA [0,00) lim f (x)dx noo (12) Suppose that f: [0, o0) - (0, 00) and that f e R((0, n]), for every n E N. Prove that f is Lebesgue measurable, the Lebesgue integral Jo.0)f dA exists, and f dA [0,00) lim f (x)dx noo
1 00 0 1 0 00 -2 3 0 0 0 1 I = 0 0 0 0 6. (10%) Let matrices A and 0 -4 5 0 1 0 -6 7 0 0 0 1 B=(I+A) (I-A) , please calculate the matrix (I+ B) - o0 1 00 0 1 0 00 -2 3 0 0 0 1 I = 0 0 0 0 6. (10%) Let matrices A and 0 -4 5 0 1 0 -6 7 0...
b) (10 pts) Let D(0, oo)) be the vector space of all bounded continuous functions from [0, oo) such that R If(x) dz 00. Give an example of a sequence {fn} of functions in D(0,00)) which (i) converges pointwise for E [0, oo) to the constant function f(z)0 (ii) does not converge to 0, neither with respect to the norm, nor the Hint: it may be helpful to contemplate the phrase "mass escaping to infinity". norm. b) (10 pts) Let...
4A. Solve the all pairs shortest path problem for the graph indicated by the weight matrix 5 in Fig. Q.4A 0 2 o0 1 8 6 0 3 2 00 00 00 0 4 00 oo 00 2 0 3 3 o0 00 00 0 Fig. Q.4A
Let f be a real-valued continuous function on R with f (-o0 0. Prove that if f(xo) > 0 for some zo R, then f has the maximum on R, that is, there exists an M R such that f(x) < f(xM) for al E R. Let f be a real-valued continuous function on R with f (-o0 0. Prove that if f(xo) > 0 for some zo R, then f has the maximum on R, that is, there exists...
Finish the proof of Theorem 3.14. Theorem 3.14 Let (neN aand EneN be sequences in R. Let be in R# and suppose that x" → x, y, → oo, and z" →-oo. . If -oo <x o, then +yn 2. If-oo x < 00, then x" + Zn →-00 4. If-oo x < 0, then xoY" →-00 and xnZn → oo. 5. If x is in R. then-→0and-" →0 Proof Note that the conditions in the different parts of the...
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...
TRIAL 1 TRIAL 2 TRIAL 3 TRIAL 4 15-O0 25.00 15.95.00 IS, O0 25.00 IS.00 25.00 Volume of HCl used (mL) Mean molarity of NaOH (M, four sig. figs!) 0.O68amo.0amo.o680.Ocbam 7.95m.30m24.90mL 0.15 mL Vinitial (mL, two decimals!) 18-10mL 23.65m 3a.30mL Vfinal (mL, two decimals!) 5600MC 4.1mL VNaOH used (mL, two decimals!)17.95mL I5.omL 13. ImL MNaeH (show calculations on back) HCI