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numerical analysis

Questions and Answers of Numerical Analysis

Decide which of the following statements are true and which are false. Prove the true ones and give counterexamples to the false ones.a) Suppose that E is a set. If there exists a function f from E
Suppose that A and B are sets and that B is uncountable. If there exists a function which takes A onto B, prove that A is uncountable.
Suppose that A is finite and f is 1-1 from A onto B. Prove that is finite.
Let f : A → and g : B → C and define g o f : A → C by (g o f)(x) := g(f(x)). a) Show that if f, g are 1-1 (respectively, onto), then g o f is 1-1 (respectively, onto). b) Prove that if f is 1-1
Suppose that n ∈ N and ϕ : {1. 2,..., n) → {1, 2 «}. a) Prove that ϕ is 1-1 if and only if ϕ is onto. b) [PIGEONHOLE PRINCIPLE] Suppose that E is a finite set and that f : E → E. Prove that
A number x0 ∈ R is called algebraic of degree n if it is the root of a polynomial P(x) = anxn + ............... + a1x + a0, where aj ∈ Z, an ≠ 0, and n is minimal. A number x0 that is not
Decide which of the following statements are true and which are false. Prove the true ones and provide a counterexample for the false ones. a) If xn converges, then xn/n also converges. b) If xn does
Using the method of Example 2.2i, prove that the following limits exist. a) 2 - 1/n → 2 as n → ∞. b) 1 + π/√n → 1 as n → ∞. c) 3(1 + 1/n) → 3 as n → ∞. d) (2n2 + l)/(3n2) → 2/3
Suppose that xn is a sequence of real numbers that converges to 1 as n → ∞. Using Definition 2.1, prove that each of the following limits exists. a) 1 + 2xn → 3 as n → ∞. b) (πxn - 2)/xn
For each of the following sequences, find two convergent subsequences that have different limits. a) 3-(-l)n b) (-l)3n + 2 c) (n - (-1)nn - 1/n
Suppose that xn ∊ R. a) Prove that {xn} is bounded if and only if there is a C > 0 such that |xn| < C for all n ∊ N. b) Suppose that {xn} is bounded. Prove that xn/nk → 0, as n → ∞, for all
Let C be a fixed, positive constant. If {bn} is a sequence of nonnegative numbers that converges to 0, and {xn} is a real sequence that satisfies |xn - a| < Cbn for large n, prove that xn converges
a) Suppose that {xn) and {yn} converge to the same real number. Prove that xn - yn → 0 as n → ∞. b) Prove that the sequence {n} does not converge. c) Show that there exist unbounded sequences
Determine which of the following statements are true and which are false. Prove the true ones and provide counterexamples for the false ones. a) If xn → ∞ and yn → -∞, then xn + yn → 0 as n
Prove that each of the following sequences converges to zero. a) xn = sin(log n + n5 + en2)/n b) xn = 2n/(n2 + π) c) xn = (√2n + l)/(n + √2) d) xn = n/2n
Use Definition 2.14 to prove that each of the following sequences diverges to + ∞ or to - ∞. a) xn = n2 - n b) xn = n - 3n2 c) xn = n2+ 1/n d) xn = n2(2 + sin(n3 + n + 1))
Find the limit (if it exists) of each of the following sequences. a) xn = (2 + 3n - 4n2)/(1 - 2n + 3n2) b) xn = (n3 + n - 2)/(2n3 + n - 2) c) xn = √3n + 2 - √n d) xn = (√4n + 1 - √n -
a) Prove Theorem 2.12iv. b) Prove Corollary 2.16.
Suppose that x ∊ R, xn > 0, and xn → x as n → ∞. Prove that √xn → √x as n → ∞. [For the case x = 0, use inequality (8) in Section 1.2.]
Prove that given x ∊ R there is a sequence rn ∊ Q such that rn → x as n → ∞.
Suppose that x and y are extended real numbers and that {xn}, {yn}, and {wn} are real sequences. a) [SQUEEZE THEOREM FOR ]. If xn → x and yn → x, as n → ∞, and xn < wn < yn for n ∊ N,
Using the result in Exercise 2.2.5, prove the following results. a) Suppose that 0 < x1 < 1 and xn+1 = 1 - √1 - xn for n ∊ N. If xn → x as n → ∞, then x = 0 or 1. b) Suppose that x1 > 3 and
a) Suppose that 0 < y < 1/10n for some integer n > 0. Prove that there is an integer 0 < w < 9 such that w/10n+1 < y < w/10n+1 + 1/10n+1.b) Prove that given x ∊ [0, 1) there
Decide which of the following statements are true and which are false. Prove the true ones and provide counterexamples for the false ones.a) If xn is strictly decreasing and 0 < xn < 1/2, then
Suppose that x0 ∊ (-1, 0) and xn = √xn-1 + 1 - 1 for n ∊ N. Prove that xn ↑ 0 as n → ∞. What happens when x0 ∊ [- 1, 0]?
Suppose that x0 = 2\/3, y0 = 3,and yn = √xnyn–1 for n e N.a) Prove that xn ↓ x and yn ↑ y, as n → ∞, for some x, y ∊ R.b) Prove that x = y and 3.14155 < x < 3.14161.(The
Suppose that 0 < x1 < 1 and xn+1 = 1 - √1 - xn for n e N. Prove that xn ↓ 0 as n → ∞ and xn+1/xn → 1/2, as n → ∞.
Suppose that x0 > 2 and xn = 2 + √xn-1 - 2 for n ∊ N. Use the Monotone Convergence theorem to prove that either xn → 2 or xn → 3 as n → ∞.
Suppose that x0 ∊ R and xn = (1 + xn-1)/2 for n ∊ N. Use ↑he Monotone Convergence theorem to prove that xn → 1 as n → ∞.
a) Suppose that {xn} is a monotone increasing sequence in R (not necessarily bounded above). Prove that there is an extended real number x such that xn → x as n → ∞. b) State and prove an
Suppose that E ⊂ R is a nonempty bounded set and that sup E ∉ E. Prove that there exists a strictly increasing sequence {xn} that converges to sup E such that xn ∊ E for all n ∊ N.
Let 0 < y1 < x1 and seta) Prove that 0 < yn < xn for all n e N.b) Prove that yn is increasing and bounded above, and that xn is decreasing and bounded below.c) Prove that 0 < xn+1 –
Suppose that x0 = 1, y0 = 0, xn = xn-1 + 2yn-1, and yn = xn-1 + yn-1 for n ∊ N. Prove that x2n - 2y2n = ± 1 for n ∊ N and xn/yn → √2 as n → ∞
Decide which of the following statements are true and which are false. Prove the true ones and provide a counterexample for the false ones. a) If {xn} is Cauchy and {yn} is bounded, then {xnyn} is
Prove that if {xn} is a sequence that satisfiesfor all w ˆŠ N, then {x"} is Cauchy.
Suppose that xn ∊ Z for n e N. If {x"} is Cauchy, prove that xn is eventually constant; that is, that there exist numbers a ∊ Z and N ∊ N such that xn = a for all n > N.
Suppose that xn and vn are Cauchy sequences in R and that a ∊ R. a) Without using Theorem 2.29, prove that axn is Cauchy. b) Without using Theorem 2.29, prove that xn + yn is Cauchy. c) Without
Let {xn} be a sequence of real numbers. Suppose that for each Îµ > 0 there is an N ˆŠ N such that m > n > N implies |ˆ‘mk=n xk|exists and is finite.
Prove that limn→∞ (- l)k/k exists and is finite.
Let {xn} be a sequence. Suppose that there is an a ∊ (0, 1) such that |xn+1 - xn| < an for all n∊N. Prove that xn → x for some x ∊ R.
a) Let E be a subset of R. A point a ∊ R is called a cluster point of E if E ∩ (a - r, a + r) contains infinitely many points for every r > 0. Prove that a is a cluster point of E if and only if
a) A subset E of R is said to be sequentially compact if and only if every sequence xn ∊ E has a convergent subsequence whose limit belongs to E. Prove that every closed bounded interval is
Find the limit infimum and the limit supremum of each of the following sequences. a) xn = 3-(-1)n b) xn = cos {nπ/2) c) xn = (-l)n+1 + (-!)n/n d) xn = √1 + n2/(2n - 5) e) xn = yn/n, where {yn} is
Suppose that {xn} is a real sequence. Prove thatand
Let {xn} be a real sequence and r ˆŠ R.a) Prove thatfor n large,b) Prove thatfor infinitely many n ˆŠ N.
Suppose that {xn} and {yn} are real sequences.a) Prove thatprovided that none of these sums is of the form ˆž - ˆž.b) Show that if limn†’ˆž xn exists, thenandc) Show by examples that each
Let {xn} and {yn} be real sequences.a) Suppose that xn > 0 and yn > 0 for each n ˆŠ N. Prove thatprovided that the product on the right is not of the form 0 €¢ ˆž. Show by example that
Suppose that xn > 0 and yn > 0 for all n ˆŠ N. Prove that if xn †’ x as n †’ ˆž (x may be an extended real number), thenprovided that none of these products is of the form 0 €¢
Prove thatfor any real sequence {xn}.
Suppose that xn > 0 for n ˆŠ N. Under the interpretation 1/0 = ˆž and 1/ˆž = 0, prove that
Let xn ˆŠ R. Prove that xn †’ 0 as n †’ ˆž if and only if
Let a ∊ R and let f and g be real functions defined at all points x in some open interval containing a except possibly at x = a. Decide which of the following statements are true and which are
Using Definition 3.1, prove that each of the following limits exists.a)b)c)d)
Decide which of the following limits exist and which do not. Prove that your answer is correct. (You can use well-known facts about the values of tan x, cos x, and log x, e.g., that log x †’
Evaluate the following limits using results from this section. (You may assume that sin x, 1 - cos x, tan x, and 3ˆšx converge to 0 as x †’ 0.)a)b)c)d)e)
Suppose that f is a real function.a) Prove that ifexists, then |f(x)| †’ |L| as x †’ a.b) Show that there is a function such that, as x †’ a, |f(x)| †’ |L| but the limit of f (x) does
For each real function f, define the positive part of f byand the negative part of / bya) Prove that f+(x) > 0, f-(x) > 0, f(x) = f+(x) - f-(x% and |f(x)| = f+(x) + f-(x) all hold for every x
Let f, g be real functions j and for each x ˆŠ Dom (f) ˆ© Dom (g) define(f ‹ g)(x) :=max{f(x),g(x)} and (f ‹€ g)(x) := mm{f (x), g(x)}.a) Prove thatandfor all x ˆŠ Dom (f) ˆ© Dom
Suppose that a ∊ R and / is an open interval which contains a. If f : I → R satisfies f(x) → f(a), as x → a, and if there exist numbers M and m such that m < f(a) < M, prove that there exist
Decide which of the following statements are true and which are false. Prove the true ones and provide counterexamples for the false ones.a) If f(x) †’ ˆž as x †’ ˆž and g(x) > 0, then
For each of the following, use definitions (rather than limit theorems) to prove that the limit exists. Identify the limit in each case.a)b)c)d)e)
Assuming that ex †’ ea, sin x †’ sin a, and cos x †’ cos a as x †’ a for any aˆŠR, evaluate the following limits when they exist.a)b)c)d)e)f.
Recall that a polynomial of degree n is a function of the formP{x) =anxn + an-1 + ˆ™ ˆ™ ˆ™ ˆ™ + a1x + a0,where aj ˆŠ R for j = 0,1,..., n and an ‰ 0.a) Prove that if 00 = 1,
Prove the following comparison theorems for real functions f and g, and a ˆŠ R.a) If f(JC) > g(x) and g(x) †’ ˆž as x †’ a, then f(x) †’ ˆž as x †’ a.b) If f(x)then g(x)
Prove the following special case of Theorem 3.17: Suppose that f : [a, ∞) → R for some a ∊ R. Then f(x) → L as x → ∞ if and only if f(xn) → L for any sequence xn ∊ (a, ∞) which
Suppose that f : [0, 1] R and f(a) = limx→a f(x) for all a ∊ [0,1]. Prove that f(q) = 0 for all q ∊ Q ∩ [0, 1] if and only if f(x) = 0 for all x ∊ [0, I].
Suppose that P is a polynomial and that P(a) > 0 for a fixed a ˆŠ R. Prove that P(x)/(x - a) †’ ˆž as x †’ a+, P(x)/(x - a) †’ -ˆž as x †’ a-, butdoes not exist.
Suppose that f: N †’ R. Ifprove that limn†’ˆž f(n)/n exists and equals L.
Decide which of the following statements are true and which are false. Prove the true ones and provide counterexamples for the false ones. a) If f is continuous on [a, b] and J := f([a, b]), then J
Use limit theorems to show that the following functions are continuous on [0,1].a)b)c)d)
If f : R †’ R is continuous andprove that f has a minimum on R; that is, there is an xm ˆŠ R such that
Let a > 1. Assume that ap+q = apaq and (ap)q = apq for all p,q ˆŠ Q, and that ap For each x ˆŠ R, definea) Prove that A(x) exists and is finite for all x ˆŠ R, and that A(p) = ap for all p
For each of the following, prove that there is at least one x ∊ R which satisfies the given equation. a) ex = x3 b) ex = 2cos x + 1 c) 2x = 2 - 3x
If f : [a, b] → [a, b] is continuous, then f has a fixed point; that is, there is a c ∊ [a, b] such that f(c) = c.
Show that there exist nowhere-continuous functions f and g whose sum f + g is continuous on R. Show that the same is true for the product of functions.
Suppose that f : R → R satisfies f(x + y) = f(x) + f(y) for each x, y ∊ R. a) Show that f(nx) = nf(x) for all x ∊ R and n ∊ Z. b) Prove that f(qx) = qf(x) for all x e R and q ∊ Q. c) Prove
Suppose that f : R → (0, ∞) satisfies f(x + y) = f(x)/(y). Modifying the outline in Exercise 3.3.8, show that if f is continuous at 0, then there is an a ∊ (0, ∞) such that f(x) = ax for all
Decide which of the following statements are true and which are false. Prove the true ones and provide counterexamples for the false ones. a) If f is uniformly continuous on (0, ∞) and g is
Using Definition 3.35, prove that each of the following functions is uniformly continuous on (0,1). a) f(x) = x2 +x b) f(x) = x3 - x + 2 c) f(*) = x sin 2x
Prove that each of the following functions is uniformly continuous on (0, 1). (You may use l'Hopital's Rule and assume that sin x and log x are continuous on their domains.)a)b)c) f(x) = x log xd)
Assuming that sin x is continuous on R, find all real a such that xα sin(l/x) is uniformly continuous on the open interval (0, 1).
a) Suppose that f : [0, ∞) → R is continuous and that there is an L ∊ R such that f(x) → L as x → ∞. Prove that / is uniformly continuous on [0, ∞). b) Prove that f(x) = l/(x2 + 1) is
Suppose that a ∊ R, that £ is a nonempty subset of R, and that f, g : E → R are uniformly continuous on E. a) Prove that f + g and αf are uniformly continuous on E. b) Suppose that f, g are
a) Let I be a bounded interval. Prove that if f : I → R is uniformly continuous on I, then f is bounded on I. b) Prove that a) may be false if I is unbounded or if I is merely continuous.
Suppose that f is continuous on [a, b]. Prove that given ε > 0 there exist points x0 = a < x1 < ∙ ∙ ∙ < xn = b such that if Ek: = {y : f(x) = y for some x ∊ [xk-1, xk]}, then sup Ek - inf
Let ⊂ R. A function f : E → R is said to be increasing on E if and only if x1, x2 ∊ E and x1 < x2 imply f(x1) < f(x2). Suppose that f is increasing and bounded on an open, bounded, nonempty
Prove that a polynomial of degree n is uniformly continuous on R if and only if n = 0 or 1.
Suppose that f,g : [a, b] → R. Decide which of the following statements are true and which are false. Prove the true ones and provide counterexamples for the false ones. a) If f = g2 and / is
For each of the following real functions, use Definition 4.1 directly to prove that f'(a) exists. a) f(x) = x2 + x, ∊ R b) f{x) = √x, a > 0 c) f(x) = l/x, a ≠ 0
a) Prove that (x")' = nxn-1 for every n eN and every x e R. b) Prove that (xn")' = nxn-1 for every n ∊ -NU{0} and every x ∊ (0, ∞).
Suppose thatShow that fa{x) is continuous at x = 0 when a > 0 and differentiable at x = 0 when a > 1. Graph these functions for a = 1 and a = 2 and give a geometric interpretation of your
Let I be an open interval which contains 0 and f : I → R. If there exists an a > 1 such that |f(x)| < |x|a for all x ∊ I, prove that f is differentiable at 0. What happens when a = 1?
a) Find all points (a, b) on the curve C, given by y = x + sin x, so that the tangent lines to C at (a, b) are parallel to the line y = x + 15. b) Find all points (a, b) on the curve C, given by v =
Define f on R byFind all n ˆŠ N such that f(n) exists on all of R.
Suppose that f : (0, ∞) → R satisfies f(x) - f(y) = f(x/y) for all x, y ∊ (0, ∞) and f(l) = 0. a) Prove that f is continuous on (0, ∞) if and only if f is continuous at l. b) Prove that f
Let I be an open interval, f : I †’ R, and c ˆŠ l. The function f is said to have a local maximum at c if and only if there is a Î´ > 0 such that f(c) > f(x) holds for all |x - c| a) If
Suppose that I = (-a, a) for some a > 0. A function f : I → R is said to be even if and only if f(-x) = f(x) for all x ∊ I, and said to be odd if and only if f(-x) = - f(x) for all x ∊ I. a)
Suppose that I is an open interval containing a, and that f, g, h : I †’ R. Decide which of the following statements are true and which are false. Prove the true ones and provide counterexamples
Suppose that f and g are differentiable at 2 and 3 with f'{2) = a, f'(3) = ft, g'(2) = c, and g'(3) = d. ft. If f(2) = 1, f(3) = 2, g(2) = 3, and g(3) = 4, evaluate each of the following
Suppose that f is differentiable at 2 and 4 with f(2) = 2, f(4) = 3. f'(2) = π, and f'(4) = e. a) If g(x) = xf(x2), find the value of g'(2). b) If g(x) = f2(√x), find the value of g'(4). c) If

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