Let (Jn)neN be a sequence of functions f: R → R satisfying |fn+1(c) – fn(c)|< a\fn(c) – fn-1(c)[for n > 2 and neN , and ae(0, 1). Prove that (Sn)n€N is a pointwise convergent sequence of functions at x = c.

Q: Derive and use the estimate 800 |u (x, t)| < 1 -7²t/10 re (2k + 1)* k=0

A:

Q: Let f(z) = 4r- 12x +7 then f(r) is decreasing on Select one: O a (-00, -1), [12, o0) Ob [1,0) O…

A:

Q: Evaluate the function f(x) = e* + In (x + 7) at c, where –7 < c < 6.

A: Given information : f(x)=ex+ln(x+7) 'c' lies between -7 and 6

Q: Evaluate the upper and lower sums for f(x) = 2 + sin x, 0<x< T, with n = 2, 4, and 8.

A:

Q: Let f(x)=0 for x<1 and f(x)=(x−1)2 for x≥1. Find the smallest integer n such that f(n)(1) does not…

A: Ans:  for     n≥2 f(n)(1) does not exists.

Q: 1 h(s) = (s – 10)2 Find the largest value of A such that the function h(s) is increasing for all s…

A: Given: h(s) = 1s-102  Increasing interval :(-∞ , A ). The function f(x) is increasing on an interval…

Q: Consider the sequence of functions {fn(x)} = {n²x(1 – x)"} for x € R and n = 1, 2,3.... (a) Sketch…

A: The given sequence: {fn(x)}={n2x(1-x)n} for x∈ℝ and n=1,2,3,... Choose x=0.5∈ℝ.   a) Sketch on the…

Q: Let f(r) = 2+ 3r – z. Find (a) the intervals on which f is increasing, (b) the intervals on which f…

A: Topic- application of derivatives

Q: Find an example of fn → f in measure, but for every x, fn(x) does not converge to f(x).

A: We have to find an example of fn→f in measure, but for every x, fn(x) does not converge to f(x).

Q: fn. (1,00) Evaluate lim

A:

Q: Consider the function f: N→ N given recursively by f(0) = 2 and f(n + 1) = 3. f(n). Find f(11)…

A: Given that f(0)=2f(n+1)=3f(n)f(1)=3f(0)=3*2=6f(2)=3f(1)=3*6=18f(3)=3f(2)=3*18=54and so on…

Q: Let t represent the total number of hours that a truck driver spends during a year driving on a…

A: Probability of at least one accident = p(t) Probability of no accident = 1 - p(t)

Q: Suppose that f'(t)-29 (b)31 (e)none of these

A: Given: Suppose that f'(t)&lt;3 for all t in R and f(3)=1. To find: f(-7)  Lagrange’s mean value…

Q: Let f: X →Y be an injective function, and A, B C X. Prove that Q1) f(A)n f(B) C f(AN B).

A:

Q: Let f(x) = xell. Determine the intervals of concavity of f(x). f(x) is concave up on (-∞, -1) U (0,…

A:

Q: (c) A non-constant function f : [0, 1] → R such that f(x) > 0 for any r e [0, 1] but | f(x)dx = 0.…

A: Let us consider a function f:0,1→ℝ such that fx≥0 for all  x∈0,1.  Suppose fxn be a sequence of…

Q: Evaluate L{f(t)} for if 0 < t,t + 1,t # 2 if t = 1 1 (i) f(t) = 3 -3 ift = 2 (ii) f(t) = 1 2t +1 if…

A:

Q: (b) Let I be an interval and f: IR be increasing. Show that fn(x) = f(x) +, defined on I, is…

A:

Q: Let Calculate f(x) = ² Σ K=O k f(K) 2 k! where K=0,1₁.. X>0

A:

Q: Let fn(x) x + n a. Show that fn(x) converges pointwise [0, 0). on

A:

Q: Let f(x) = (x – 1)2 + (x + 1)³. Find the sum. f(1) + f(2) + f(3) + f(4).

A: Givenf(x)=x-12+x+13

Q: Let x1 O and xn+1 x+1 2 Show that (x,n) converges and find its limit.

A: It is given that  x1=0 and xn+1=xn2+12. Let xn+1=ϕ(xn)=xn2+12 By fixed point iteration method…

Q: Let f(t) = 2t² for -4 < t <4 and extend periodically. Compute: f(1) = f(5) = f(-4.5) = f(-8)= f(12)…

A: Periodic function: A function is said to be periodic with period T if f(x+T)=f(x).

Q: Suppose that the function f:R → R be differentiable and that {xn}is a strictly increasing bounded…

A:

Q: Let f(x) = In(x) x3 O (0, e¹/³) 0 (-∞, ∞) O (e¹/3, ∞) ○ (0, ∞) O (-∞, e¹/3) Find the intervals where…

A: I am attaching image so that you understand each and every step.

Q: 2x is an increasing function for x>0. 2+x Show that y = In (1+x)-

A:

Q: 14. Let {f} be a sequence on C(X) where X CR is compact. Assume that fn (x) ≥ fn+1(x) for all x E X.…

A: Given:A sequence of functions  in the space of complex-valued continuous functions , where  is a…

Q: (b) Consider the set U = {(x, y) E R² : 2x + y < 1, x 2 0, y 2 0} and let Zn = (Xn, yn), where Xn =…

A: NOTE:- I HAVE REQUIRED QUESTION A TO SOLVE QUESTION C AND D. THEREFORE I AM SOLVING ONLY QUESTION B.

Q: Q2/ Let x(n) be a 4-point sequence. Compute the DFT of x(n). 1 x (n) =- 0<n<3 0otherwise ans. X(k)…

A: …

Q: Q1 Untitled - Notepad Let a, a2, a3, .. be the sequence of numbers defined by a, = 1, a, = /2 + a, 1…

A: We prove our required result by induction.

Q: Suppose fn → f and the functions fn all satisfy the Lipschit condition |fn(x) – fn(y)| < Mx – y| for…

A:

Q: (b) Let the function h : Z12 → Z12 be given by the rule h([x]) = [3] © [x]. (i) Compute h([5]). (ii)…

A: