5. Consider the family F = {sin(nx)}_₁ in Cº([0, π]). (a) Is bounded? (b) Is Fequicontinuous? (c) Does have a convergent subsequence? (d) Is compact?

Q: Suppose g is a continuous DECREASING function on the interval [-3,1].

A: 2. As y is a continous decreasing function on the interval -3, 1

Q: 3. Let f be continuous on [-1, 4] and differentiable on (-1,4). If f(-1) = –1 and f(4) = 4, then the…

A:

Q: f(x) is continuous on (- 00,00). Use the given information to sketch the graph of f. f(0) = 14, f(7)…

A: Given ,  f(x) is continuous on (-∞, ∞)  We have to use the given information to sketch the graph of…

Q: d. y= x² + y x + cos y

A:

Q: Suppose the function f(x) is differentiable on (0, ∞) with a continuous derivative and satisfies >…

A: Step 1: Step 2: Step 3:

Q: sin (xy). In x dx Finding the uniform convergence of the integral

A: Given: ∫01sinxylnxxdx, y≤1 By using the infinite series expansion of  sinθ=∑n=0∞-1n2n+1ǃθ2n+1, the…

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: 0 (n = 0, 1, 2, ...). When f is continuous (10 points) Define f (x) = x" sin (1/x) for x everywhere?…

A: Let's define the function f(x) = x^n * sin(1/x) for x ≠ 0 and f(0) = 0. This function is continuous…

Q: (6) Decide if the following improper integrals are convergent or divergent using Basic Comparison…

A: Calculating convergence of given improper integrals using basic comparision test

Q: Rolle's Theorem states: Let f be continuous on the closed interval [a, b] and differentiable on the…

A:

Q: Let fn(x) on [0, 1]. 1+ x" (a) Prove that fn converges uniformly to 0 on [0, €], Ve E (0, 1). (b)…

A: According to our guideline, we can solve only first question of multiple question. If you want…

Q: If f is continuous on [0, π], use the substitution u = π-x to show that π [" xf(sin(x)) dx =…

A: To prove

Q: If f is continuous on [0, π], use the substitution u = -x to show that -76 2 ["xf(s xf(sin(x)) dx =…

A:

Q: 3c) The integrand is continuous in its domain (0, +∞o). Therefore, we study the be- haviour of the…

A:

Q: Assume Y is continuous on [c, d] and differentiable on (c,d). (a) If y is non-decreasing (monotone…

A: We have used the definition of differentiability to prove 1st part. And given a counter example in…

Q: Σ sin k+1

A: to show that the given series ∑k=1∞1ksinxk+1 converges uniformly on R to a differentiable function f…

Q: for z+ - and Let f(z) = for z or (a) Show that is continuous at -n/2.

A: a) When x≠-π2 limx→-π2cosxx2-π24

Q: If f' is continuous on [a, b], show that 2 [* f(x)f'(x) dx = [f(b)J* – [f(@}J°

A: Proof:            Given: f' is a continuous…