Calculate vf for the function

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Reference no: EM131081393

Math 104: Final exam-

1. Suppose that (sn) is an increasing sequence with a convergent subsequence. Prove that (sn) is a convergent sequence.

2. (a) Let fn be functions defined on R as

fn(x) = x/n2(1 + nx2),

for all n ∈ N. By using the Weierstrass M-test, or otherwise, prove that ∑ fn converges uniformly on R.

(b) Consider the sequence of functions defined on R as

1885_Figure2.png

for all n ∈ N. Define Mn = sup{|gn(x)| : x ∈ R}. Prove that ∑gn converges uniformly to a limit g, but that ∑Mn diverges. Sketch g(x) for 0 ≤ x < 5.

3. Consider the sequence (sn) defined according to s1 = 1/3 and sn+1 = λsn(1 - sn) for n ∈ N, where λ is a real constant.

(a) Prove that (sn) converges for λ = 1.

(b) Prove that (sn) has a convergent subsequence for λ = 4.

(c) Prove that (sn) diverges for λ = 12.

4. A real-valued function f on a set S is defined to be Lipschitz continuous if there exists a K > 0 such that for all x, y ∈ S,

| f(x) - f(y)| ≤ K|x - y|.

(a) Prove that if f is Lipschitz continuous on S, then it is uniformly continuous on S.

(b) Consider the function f(x) = √x on the interval [0, ∞). Prove that it is uniformly continuous, but not Lipschitz continuous.

5. Consider the function defined on the closed interval [a, b] as

1734_Figure5.png

where a < c < b.

(a) Prove that hc is integrable on [a, b] and that ab hc = 0.

(b) Suppose f is integrable on [a, b], and that g is a function on [a, b] such that f(x) = g(x) except at finitely many x in (a, b). Prove that g(x) is integrable and that ab f = ab g. You can make use of basic properties of integrable functions.

6. Consider the sequence of functions hn on R defined as

991_Figure6.png

(a) Sketch h1 and h2.

(b) Prove that hn converges pointwise to 0 on R.

(c) Let f be a real-valued function on R that is differentiable at x = 0. Prove that

limn→∞ -∞hn f = f'(0).

7. (a) Find an example of a set A ⊆ R where the interior Ao is non-empty, but that sup A ≠ sup Ao and inf A ≠ inf Ao.

(b) Let B1, . . . , Bn be subsets of R. Prove that

(i=1n Bi)o = i=1n Boi.

(c) Suppose that {Ci}i=1 is an infinite sequence of subsets of R. Prove that

(i=1 Ci)oi=1 Coi.

but that these two sets may not be equal.

8. (a) If f is a continuous strictly increasing function on R, prove that d(x, y) = |f(x) - f(y)|

defines a metric on R.

(b) Prove that d is equivalent to the Euclidean metric dE(x, y) = |x - y|.

(c) Suppose g is a continuous strictly increasing function on [0, ∞) where g(0) = 0. Is the function

d2(x, y) = g(|x - y|)

always a metric? Either prove the result, or find a counterexample.

9. Consider the continuous function defined on R as

220_Figure9.png

where c ∈ R. For this question you can assume basic properties of the exponential, such that it is continuous, differentiable, and has the Taylor series ex = k=0xk/k!.

(a) Use L'Hopital's rule to compute limx→0 f(x) and hence determine c.

(b) Show that f is differentiable on R and compute f'.

(c) Use the above results to write down the partial Taylor series

fT(x) = k=01f(k)(0)xk/k!.

Calculate the function limits limx→∞ f(x) and limx→-∞(x + f(x)) and use the results to sketch f and fT on (-10, 10).

10. Given a function f on [a, b], define the total variation of f to be

Vf = sup{k=1n|f(tk)-f(tk-1)|}

where the supremum is taken over all partitions P = {a = t0 < t1 < . . . < tn = b} of [a, b].

(a) Calculate Vf for the function defined on [-1, 1] as

2005_Figure10.png

(b) Prove that if f is differentiable on an interval [a, b] and that f' is continuous then Vf = ab|f'|.

Reference no: EM131081393

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