Let fn: R --> R be defined by : fn(x)= x/(1+nx2), For all n >= 1. a) Show that {fn} converges uniformly on R to a function f. b) Show that f'(x) = limn -->infinity f'n(x), For all x does not = 0, but this equality is false for x = 0. c)What assumption in the theorem on the interchange of the limit and the derivative is missing? I am stuck with that last part (C).

Q: Let xn -> 0 and (yn) be bounded. Prove xnyn -> 0.

A: A sequence is bounded if it is bounded above and below, that is to say, if there is a number, k,…

Q: 27. The graph of a function f is given. Let F(x)=f(1)dt = f(1)dt_then for 0 0<x<2 F(x) is: [A]…

A:

Q: Let fn, f be real functions on S2. Show ∞ ∞ ∞ {w: fn(w) + f(w)} = UNU {w: | fn(w) - f (w)| ≥ } }.…

A:

Q: Provide a function f, continuous on (0, 1), such that for all r where f'(x) exists, f'(x) > 0 but f…

A:

Q: Suppose f(x) is continuous on [3, 8] and -5 < f'(x) < 3 for all x in (3, 8). Use the Mean Value…

A:

Q: Suppose the function fis continuous and has domain (-0o, 00) and has all of the following…

A: You have posted multiple subparts. As per the instruction, we will be answering only the first four…

Q: ,10 If f is a continuous function and F'(x)=f(x) for all x € R, then A 2[F(5) – F(1)] B 2[F(10) –…

A:

Q: Suppose a certain real-valued function f is continuous on the interval [33, 50] and differentiable…

A:

Q: Define f(x)=x^3 for all x. Verify the ε-δ criterion for continuity at each point x_0.

A: X3 is continuous at every point in R.

Q: Sketch a graph of a function f that is continuous on (-∞,∞) and has the following properties. f'(x)…

A: Increasing/Decreasing Functions: The derivative of a function used to determine whether the function…

Q: Suponga que f(r) = -2 y que esta función se define en intervalo [-0.5,1]. El mínimo absoluto de…

A: Answer to the question is as follows:

Q: Which statement is true about a function f(x), where f'(x) = 3x²_²? X Of is increasing for x > 0.874…

A: Given here the function and put here

Q: Let f(x) Determine the intervals on which f'(x) > 0 and f'(x) 0

A:

Q: t fn(x) = sin^n(x) for all x [0,pi]. Prove that fn' is not uniformly convergent on [0,pi].…

A: We can prove it by showing that function is not pointwisw convergent.

Q: Show that f is a continuous function for all x, on the interval -L < x ≤ L f(x) = (x / L) (1 - |x /…

A:

Q: Let h be a differentiable function. Also he [a, b]. Prove that there is a sequence {x} = (a, b)such…

A:

Q: Suppose 10 < f'(x) < 20 for all x E R (and so f(x) is continuous and differentiable on R). Use the…

A:

Q: Consider f, a function such as f(2) = 1, f'(2) = 2, f''(2) = 3 and f'''(2) = 4 and that verifies:…

A: Given: f(2)=1, f'(2)=2, f''(2)=3 and f'''(2)=4 To find: Error bound in the approximation of f1.

Q: The graph of f is shown in the figure. 6. 4 х 14 -4 -6+ 2-

A:

Q: Suppose f" is continuous on (-00, 00). (a) If f'(-5)=0 and f"(-5)= 5, what can you say about f? ○ At…

A: Q(a)At x=-5, f has a local minimum.Q(b)More information is needed to determine if f has a maximum or…

Q: Use the MVT to prove that if f' (x) ≤ 2 for x &gt; 0 and f (0) = 4, then f (x) ≤ 2x + 4 for all x ≥…

A:

Q: Given the graph of ƒ' (x) below, where the Dom f (x) = (-∞, ∞) . Find ALL interval(s) for which f""…

A:

Q: Let f (x) = + 2. 5. 131 - 4. 3. 2. On of the theit f(x) following which Interval concate dowr

A:

Q: prove that P(X)=P(X|F)P(F) + P(X|F')P(F')

A:

Q: Suppose fis a real valued function defined on an open interval I and differentiable at every x eI.…

A:

Q: f(x) is continuous on (-∞,00). Use the given information to sketch the graph of f. f'(x) f''(x) + +…

A: We have to plot the graph.

Q: QI Find f'(2) for (x³ – x)(x² + 2)(x² + x - 1) x + 2 f(x) = 587/3 O 379/7 585/4 O 676/5 O 699/7…

A: Differentiation: The process of finding the derivative of a function is called differentiation.…

Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence.  It was invented by German mathematician Carl Friedrich Gauss.