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using this as an example 1③ Consider the functions ' f(x) = {x , if xeQ...
Question 12 11. Show that if F is continuous on Rn and F(X + Y) = F(X) + F(Y) for all X : in R", then A is linear. HINT: The rational numbers are dense in the reals. 12. Find F and JF. Then find an affine transformation G such that F(X)-G(Y) lim =0. T x2+y+2z (a) F(x, y,z)coxy. Xo- (1,-1,0) e*yz ex cos y (b) Fe*sin y 1, xo=(0, π/2) 13. Find F. g1 (x) 11. Show that if...
1. Given f(x) = et XER g(x) = 3lnx x > 0,x & IR a) Find an expression for fg(x), simplifying your answer [2] b) Show that there is only one real value of x for which fg(x) = gf(x) (6) 8 Marks
5. Suppose and g are two functions continuous at x = 1. If lim and lim g(x) = 10, find f(1) and g(1). Justify your solution. mous at z = 1. If bug (in (52) s(e) * - ) = 21 6. Show that each of the following equations has a solution: a) sin(a) + x3 = 0. b) In(a)e* + 1*22 = 0).
1. Consider the function -F5 sin(r) for r f(x) =2 for 1< 3 2-25 for 3 x2 -9x + 20 Evaluate the following limits You do not have to cite limit laws, but you must show how you arrived at your answer If a limit Does Not Exist, explain why. You should use oo or -oo where applicable Calculating the limit using L'Hopital's Rule will receive NO CREDIT. (a) lim f(x) r-+0 (b) lim f(x)= z-1 (e) lim f(z) (d)...
(d) The function f(x)1 is locally integrable on (0, oo). To see whether converges, we consider the improper integrals separately. (The choice of π above is arbitrary.) By considering f (x) lim an show that 11 converges iff p< 1. Next, by considering lim J(z) an -p- dx show that /2 converges iff p +q>1. Finally, combine these results to show that I converges iff p < 1 and p+q1. (d) The function f(x)1 is locally integrable on (0, oo)....
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...
Compute the first three non-zero terms of the Taylor series for the functions: Q.1 [10 Marks] Compute the first three non-zero terms of the Taylor series for the functions: (a) (i) f(x)-In( 1 ) about a-0 where Ir < 1 (Hint: In(it)-In(1+z)-In(1-r)) (ii) From your result in (i) find ËIn(쁩) dt Page: 1 of3 MAT1841 Assignment 2 2019 Continuous Mathematics for Computer Science 3 +3+4-10 (c) h(z) = exp (sin r) about a = 픔 Q.1 [10 Marks] Compute the...
2. Consider the vector space C([0, 1]) consisting of all continuous functions f: [0,1]-R with the weighted inner product, (f.g)-f(x) g(x) x dr. (a) Let Po(z) = 1, Pi(z) = x-2, and P2(x) = x2-6r + 흡 Show that {Po, pi,r) are orthogonal with respect to this inner product b) Use Gram-Schmidt on f(x)3 to find a polynomial pa(r) which is orthogonal to each of po P1 P2 You may use your favorite web site or software to calculate the...
Problem 1. Consider the function f(x)- 3.12 show that f is Riemann integrable on [0.2] and use the definition to find .后f(x)dr Problem 2. Consider the function -2, zEQ 2, O f(r) = Show that f is not Riemann integrable on 0,1 but s Reemann integrable on this interval. Problem 3. (a) Let f be a real-valued function on a, b] such thatf()0 for all c, where c E [a, b Prove that f is Riemann integrable on a, b...
real analysis 1,2,3,4,8please 5.1.5a Thus iff: I→R is differentiable on n E N. is differentiable on / with g'(e) ()ain tained from Theorem 5.1.5(b) using mathematical induction, TOu the interal 1i then by the cho 174 Chapter s Differentiation ■ EXERCISES 5.1 the definition to find the derivative of each of the following functions. I. Use r+ 1 2. "Prove that for all integers n, O if n is negative). 3. "a. Prove that (cosx)--sinx. -- b. Find the derivative...