Let [a, b] be an inter a space. Let ƒ:[a,b] → X
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A: To Solve: the complex quadratic equation z2-(3+5j)z+8j-5=0.
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Q: B/Find the oblique asymptote of the function f(x) -3x²+2 x-1
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A: This is a problem of Fourier expansion.
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Q: Evaluate the following: 25² a. £-1 : + b. £-1 [ s²+25 4-2s c. £1[ | + 4=s? s4+8s +16 353
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Q: 2. (Calculus) Evaluate the integral S 1²+2x+3 (x-6)(x²+4) dx
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Q: 1. Let G be any group. Show that the function f: GG defined by f(x) = x² is group homomorphism if…
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- If a functionfis increasing on (a,b) and decreasing on (b,c) , then what can be said about the local extremum offon (a,c) ?Let X be a Banach space an {xn} be a sequence in. X. Provethat if {xn} converges in norm in X, then it converges weakly to thesame limitProve that if a sequence of continuous functions fn :R→R is uniformly convergent on Q, then it is uniformly convergent on R.
- Suppose that a sequence of differentiable functions {fn} converges pointwiseto a function f on an interval [a,b], and the sequence {f′n}converges uniformlyto a function g on [a,b]. Then show that f is differentiable and f′(x) = g(x)on [a,b].Let (a) Find the pointwise limit of (fn) for all x ∈ (0,∞).(b) Is the convergence uniform on (0,∞)?(c) Is the convergence uniform on (0, 1)?(d) Is the convergence uniform on (1,∞)?This exercise and the next explore partial converses of the Continuous Limit Theorem (Theorem 6.2.6). Assume fn → f pointwise on [a, b] and the limit function f is continuous on [a, b]. If each fn is increasing (but not necessarily continuous), show fn → f uniformly.
- Are the following statements true or false? If true give a proof, and if false give a counter-example: (a)Consider a continuous function f : (0, 1) → R and a Cauchy sequence Xn ∈ (0, 1).Then f(Xn) is also Cauchy. (b)If Xn <a and limn→∞: Xn =l, then l<a. (c) For an, bn ∈ R, consider a sequence of open intervals In = (an, bn).Let fn be a sequence of functions that are uniformly continuous on a set S. Suppose that (fn) converges uniformly to a function f on S. Prove that f is uniformly continuous on S.Let (fn) be a sequence of differentiable functions defined on the closed interval [a, b], and assume (fn ) converges uniformly on [a, b]. If thereexists a point x0 ∈ [a, b] where fn(x0) is convergent, then (fn) converges uniformly on [a, b]. Proof. Exercise 6.3.7. Combining the last two results produces a stronger version of Theorem 6.3.1.
- Let fn be a bounded sequence of functions uniformly convergetn to f. Prove that f is bounded as well. Is the claim true if we replace the assumption of uniform convergence with pintwise convergence.Let a > 0 and X1 = √a. Define the sequence Xn = √(a + Xn-1), n ≥ 1.Show that (Xn)n is convergent and determine its limit.Does the Bounded Convergence Theorem hold if m(E)<∞ but we drop the assumption that the sequence {|fn|} is uniformly bounded on E?