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An Introduction to Analysis 4th edition William R. Wade - Solutions

Using Exercise 4.1.2, prove that every polynomial belongs to C∞(R).
Suppose that f is differentiable at a and f(a) ‰ 0.a) Show that for h sufficiently small, f(a + h) ‰ 0).b) [RECIPROCAL RULE] Using Definition 4.1 directly, prove that 1/f(x) is differentiable at x = a andc) Use the Product Rule and the Reciprocal Rule to prove the Quotient Rule
Suppose that n ˆŠ N and f, g are real functions of a real variable whose nth derivatives f(n), g(n) exist at a point a. Prove Leibniz's generalization of the Product Rule:
a) Prove that if q = n/m for n ˆŠ Z and m ˆŠ N, thenfor every x, a ˆŠ (0, ˆž).b) [POWER RULE] Use Exercise 4.1.2 and part a) to prove that xq is differentiable on (0, ˆž) for every q ˆŠ Q and that (xq)' = qxq-1.
Assuming that ex is differentiable on R, prove thatis differentiable on [0, ˆž). Is f differentiable at 0?
Using elementary geometry and the definition of sin x, cos x, we can show that for every x ˆŠ Ri) | sin x| ii) sin(-x) = - sin x. cos(-x) = cos x,iii) sin2 x + cos2 x = 1, cos x = 1 - 2 sin2 (x/2),iv) sin(x ± y) = sin x cos y ± cos x sin y. Moreover, if x is measured in radians, then v) cos x =
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 and g are increasing on [a, b], then f + g is increasing on [a, b]. b) If f and g are increasing on [a, b], then fg is
Prove that each of the following inequalities holds. a) 2x + 0.7 < ex for all x > 1. b) log x < √x - 0.6 for all x > 4. c) sin2 x < 2|x| for all x ∈ R. d) 1 - sin x < ex for all x > 0.
Suppose that (a, b) is an open interval, that f: (a, b) → R is differentiable on (a, b), and that x0 ∈ (a, b) is a proper local maximum of f (see Exercise 4.1.8). a) Prove that given δ > 0, there exist x1 < x0 < x2 such that fʹ(x1) > 0, fʹ(x2) < 0, and |xj - x0| < δ for j = 1,2. b) Make and
Suppose that f : [a, b] → R is continuous and increasing. Prove that sup f(E) = f (sup E) for every nonempty set E ⊂ [a, b].
Suppose that fʹ is differentiable at every point in a closed, bounded interval [a, b]. Prove that if fʹ is increasing on (a, b), then fʹ is continuous on (a, b).
Let f be a real function and recall that an r ∈ R is called a roof of a function f if and only if f(r) = 0. Show that if f is differentiable on R, then its derivative fʹ has at least one root between any two roots of f.
Suppose that a < b are extended real numbers and that f is differentiable on (a, b). If f' is bounded on (a, b), prove that fʹ is uniformly continuous on (a, b).
Suppose that f is differentiable on R. If f(0) = 1 and |fʹ(x)| < 1 for all x ∈ R, prove that |f(x)| < |x| + 1 for all ∈ R.
Suppose that f is differentiable on (a, b), continuous on [a, b], and that f(a) = f(b) = 0. Prove that if f(c) ≠ 0 for some c ∈ (a, b), then there exist x1, x2 ∈ (a, b) such that f'(x1) is positive and fʹ(x2) is negative.
Suppose that f is continuous on [a, b] and that F(x) :=sup f ([a,x]). Prove that F is continuous on [a, b].
Suppose that f is twice differentiable on (a, b) and that there are points x1 < x2 < x3 in (a, b) such that f(x1) > f(x2) and f(x3) > f(x2). Prove that there is a point c ∈ (a, b) such that fʹʹ(c) > 0.
Suppose that f is differentiable on (0, ∞). If L = limx→∞ fʹ(x) both exist and are finite, prove that L = 0.
Decide which of the following statements are true and which are false. Prove the true ones and provide counterexamples for the false ones. a) x/log x → 0 as x → 0. b) If n ∈ N, then sin(1/x)/xn → 0 as A: → ∞. c) xlogx → 0 as x → 0+. d) If there is a β > 0 such that f'(x) > β for
Let f(x) = cos x and n ˆˆ N.a) Find the Taylor polynomial P2n: = P2nf,0.b) Prove that if x ˆˆ [-1, 1], thenc) Find an n so large that P2n approximates COS x on [-1, 1] to seven decimal places.
Prove I'Hopital's Rule for the case |B| = ∞ by first proving that g(x)/f(x) → 0 when f(x)/g(x) ±∞, as x → a.
Suppose that f and g are differentiable on an open interval I and that a ˆˆ R either belongs to I or is an endpoint of I. Suppose further that g and g' are never zero on I {a} and thatis of the form 0/0. If there is an M ˆˆ R such that |f'(x)/g'(x)|
Let f(x) = log x and n ˆˆ N.a) Find the Taylor polynomial Pn := pnf,1.b) Prove that if x ˆˆ [1, 2], thenc) Find an n so large that Pn approximates log x on [1, 2] to three decimal places.
Evaluate the following limits.a)b)c)d)e)f)g)h)
Let a > 0 and recall that (xα)ʹ = axα-1 and (log x)' = 1/x for all x > 0. a) Prove that log x < xa for x large. Prove that there exists a constant Ca such that log x < Caxa for all x ∈ [1, ∞), Ca → ∞ as α → 0+, and Ca → 0 as a → ∞. b) Obtain an analogue of part a) valid for ex
Assume that ex is differentiable on R with (ex)Ê¹ = ex.a) Show that the following function is differentiable on R with fÊ¹(0) = 0:b) Do analogous statements hold for f(n)(x) when n = 2, 3,...?
Suppose that n ∈ N is odd and f(n) exists on [a, b]. If f(k)(a) = f(k)(b) = 0 for all k = 0, 1, ...,n - 1 and f(c) ≠ 0 for some c ∈ (a, b), prove that there exist x1, x2 ∈ (a, b) such that f(n)(x1) is positive and f(n)(x2) is negative.
a) Prove that |δ + sin(δ + π)| < δ3/3! for all 0 < δ < 1. b) Prove that if |x - π| < δ < 1, then |x + sin x - π| < δ3/3!.
Decide which of the following statements are true and which are false. Prove the true ones and provide counterexamples for the false ones.a) Suppose that I ⊂ R is nonempty. If f : I → R is 1-1 and continuous, then f is strictly monotone on I.b) Suppose that I is an open interval which contains
Suppose that f and g are 1-1 and continuous on R. If f(0) = 2, g(l) = 2, f'(0) = π, and gʹ(1) = e, compute the following derivatives. a) (f-1)'(2) b) (g-1)'(2) c) (f-l ∙ g-l)ʹ(2)
Suppose that f is C1 on an interval (a, b). If f'(x0) ≠ 0 for some x0 ∈ (a, b), prove that there exist intervals I and J such that f is 1-1 from I onto J and f-1 is continuously differentiable on J.
Suppose that f is differentiable at every point in a closed, bounded interval [a, b]. Prove that if f' is 1-1 on [a, b], then f' is strictly monotone on [a, b].
Let f(x) = x2ex2, and assume that (ex)' = ex for all x ∈ R. a) Show that f-1 exists and is differentiable on (0, ∞). b) Compute (f-1)ʹ(e).
Using the Inverse Function Theorem, prove that (arcsin x)' = 1/√l - x2 for x ∈ (-1, 1) and (arctan x)' = 1/(1 + x2) for x ∈ (-∞, ∞).
Suppose that f' exists and is continuous on a nonempty, open interval (a, b) with fʹ(x) ≠ 0 for all x ∈ (a, b). a) Prove that f is 1-1 on (a, b) and takes (a, b) onto some open interval (c, d). b) Show that f-1 ∈ C1 (c, d). c) Using the function f(x) = x3, show that b) is false if the
Suppose that a := limx→∞ (1 + 1/x)x exists and is greater than 1 (see Example 4.22). Assume that ax: R → (0, ∞) is onto, continuous, strictly increasing, and satisfies axay = ax+y and (ax)y = axy for all x,y ∈ R (see Exercise 3.3.11). Let L(x) denote the inverse function of ax. a) Prove
Suppose that I is a nondegenerate interval, that f : I → R is differentiable, and that f'(x) ≠ 0 for all x ∈ I. a) Prove that f-l exists and is differentiable on f(I). b) Suppose further that I is a closed, bounded interval and that f' is continuous. Prove that (f-1)' is bounded on f(I).
Suppose that f : [a, b] → [c, d] is differentiable and onto. If f' is never zero on [a, b] and d - c > 2, prove that for every x ∈ [c, d] there exist x1 ∈ [a, b] and x2 ∈ [c, d] such that |fʹ(x1)(f-1,(x) - f-1(x2))| = 1.
Suppose that f is differentiable on a closed, bounded interval [a, b]. If f[a, 6] = [a, b] and f' is never zero on [a, b], prove that for every x ∈ [a, b] there exist x1 , x2 ∈ (a, b) such that F(x) = f'(x1)fʹ(x2)(f-1(x) - f-1(a)) + f(a).
Suppose that a a) If f is Riemann integrable on [a, b], then f is continuous on [a, b].b) If |f| is Riemann integrable on [a, b], then f is Riemann integrable on [a, b].c) For all bounded functions f : [a, b] †’ R,d) If f is continuous on [a, b) and on [b, c], then f is Riemann integrable on
For each of the following, compute U(f. P), L(f. P), and ˆ«20 f(x)dx, whereFind out whether the lower sum or the upper sum is a better approximation to the integral. Graph f and explain why this is so.a) f(x) = x3b) f(x) = 3 - x2c) f(x) = sin(x/5)
Let f be bounded on a nondegenerate interval [a, b). Prove that f is integrable on [a, b] if and only if given Îµ > 0 there is a partition PÎµ of [a, b] such that
a) Prove that for each n ˆˆ Nis a partition of [0, l ].b) Prove that a bounded function f is integrable on [0,1] ifin which case ˆ«10 f(x)dx equals I0.c) For each of the following functions, use Exercise 1.4.4 to find formulas for the upper and lower sums of f on Pn, and use them to compute the
Let E : = {1/n : n ˆˆ N}. Prove that the functionis integrable on |0, 1 ]. What is the value of ˆ«10 f(x)dx?
Suppose that a a) Prove that if f is continuous at x0 ˆˆ [a, b] and f(x0) ‰ 0, thenb) Show that if f is continuous on [a, b], then ˆ«ba |f(x)dx = 0 if and only if f(x) = 0 for all x ˆˆ [a, b].c) Does part b) hold if the absolute values are removed? If it does, prove it. If it does
Suppose that afor all c ˆˆ [a, b] if and only if f(x) = 0 for all x ˆˆ [a, b]. (Compare with Exercise 5.1.4, and notice that f need not be nonnegative here.)
Let f be integrable on [a, b] and E be a finite subset of [a, b]. Show that If g is a bounded function which satisfies g(x) = f (x) for all x ˆˆ [a, b]E, then g is integrable on [a, b] and
Let f, g be bounded on [a, b].a) Prove thatandb) Prove thatandfor a
a) If f is increasing on [a, b] and P = {x0, ..., xn] is any partition of [a, b], prove thatb) Prove that if f is monotone on [a, b], then f is integrable on [a, b].[By Theorem 4.19, f has at most countably many (i.e., relatively few) discontinuities on [a, b]. This has nothing to do with the proof
Let a < b and 0 < c < d be real numbers and f : [a, b] → [c, d]. If f is Riemann integrable on [a, b], prove that √f is Riemann integrable on [a, b]
Suppose that a a) If f and g are Riemann integrable on [a, b], then f - g is Riemann integrable on [a, b].b) If f is Riemann integrable on [a, b] and P is any polynomial on R, then P o f is Riemann integrable on [a, b].c) If f and g are nonnegative real functions on [a, b], with f continuous and g
Using the connection between integrals and area, evaluate each of the following integrals.(a)b)c)d)
Prove that if f and g are integrable on [a, b], then so are f ⋁ g and f ⋀ g (see Exercise 3.1.8).
Suppose that f : [a, b] → R. a) If f is not bounded above on [a, b], then given any partition P of [a, b] and M > 0, there exist tj ∈ [xj-1, xj] such that S(f, P, tj) > M. b) If the Riemann sums of f converge to a finite number I(f), as ||P|| → 0, then f is bounded on [a, b].
a) Suppose that a < b and n ∈ N is even. If f is continuous on [a, b] and ∫ba f(x)xndx = 0, prove that f(x) = 0 for at least one x ∈ [a, b]. b) Show that part a) might not be true if n is odd. c) Prove that part a) does hold for odd n when a > 0.
Use Taylor polynomials with three or four nonzero terms to verify the following inequalities.a)(The value of this integral is approximately 0.3102683.)b)(The value of this integral is approximately 1.4626517.)
Suppose that f : [0, ˆž) †’ [0, ˆž) is integrable on every closed interval [a,b] Š‚ [0, ˆž). Ifthen there is a function g : [0, ˆž) †’ [0, ˆž) such that F(x) = ˆ«xg(x) f(y)dy for all x ˆˆ [0, ˆž).
Prove that if f is integrable on [0,1] and Î² > 0, thenfor all a
a) Suppose that gn > 0 is a sequence of integrable functions which satisfiesShow that if f : [a, b] †’ R is integrable on [a, b], thenb) Prove that if f is integrable on [0,1], then
Suppose that f is integrable on [a, b], that x0 = a, and that xn is a sequence of numbers in [a, b] such that xn †‘ b as n †’ ˆž. Prove that
Let f be continuous on a closed, nondegenerate interval [a, b] and seta) Prove that if M > 0 and p > 0, then for every Îµ > 0 there is a nondegenerate interval / c [a, b] such thatb) Prove that
Let f : [a, b] †’ R, a = x0
Suppose that a a) If f is continuous and nonnegative on [a, b] and g : [a, b] †’ [a, b] is differentiable and increasing on [a, b], thenis increasing on [a, b].b) If f and g are differentiable on [a, b], if f' and g' are Riemann integrate on [a, b], and if f(a) = 0 but g is never zero on [a,
If f : R †’ R is continuous, find F'(x) for each of the following functions.a)b)c)d)
Suppose that ϕ is C1 on [a, b] and f is integrable on [c, d] := ϕ[a, b]. If ϕ is never zero on [a, b], prove that f o ϕ is integrable on [a, b].
Let q ∈ Q. Suppose that a < b, 0 < c < d, and that f : [a, b] → [c, d]. If f is integrable on [a, b], then prove that fq is integrable on [a, b].
For each n ˆˆ N, defineProve that an †’ 4/e
Suppose that f is nonnegative and continuous on [1, 2] and that ˆ«21xkf(x)dx = 5 + k2 for k = 0, 1, 2. Prove that each of the following statements is correct.a)(b)c)
Suppose that f is integrable on [0.5,2] and thatfor k = 0, 1,2. Compute the exact values of each of the following integrals.a)b)
Suppose that f and g are differentiable on [0, e] and that f' and g' are integrable on [0, e].a) If ˆ«e1 f(x)/xdxb)c) If 0 ˆˆ {f(0), g(0)} ˆ© {f(e), g(e)} prove that
Use the First Mean Value Theorem for Integrals to prove the following version of the Mean Value Theorem for Derivatives. If ∈ C1[a, b], then there is an x0 ∈ [a, b] such that f(b) - f(a) = (b - a)f'(x0).
If f is continuous on [a, b] and there exist numbers a ‰ Î² such thatholds for all c ˆˆ (a, b), prove that f(x) = 0 for all x ˆˆ [a, b].
Define L : (0, ˆž) †’ R bya) Prove that L is differentiable and strictly increasing on (0, ˆž), with L'(x) = 1/x and L(l) = 0.b) Prove that L(x) †’ ˆž as x †’ ˆž and L(x) †’ -ˆž as x †’ 0+. (You may wish to provefor all n ˆˆ N.)c) Using the fact that {xq)' = qxq-1
Let E = L-1 represent the inverse function of L, where L is defined in Exercise 5.3.7.a) Use the Inverse Function Theorem to show that E is differentiable and strictly increasing on R with E'(x) = E (x), E(0) = 1, and E (1) = e.b) Prove that E(x) †’ ˆž as x †’ ˆž and E(x) †’ 0 as x
Suppose that f : [a, b] †’ R is continuously differentiable and l-l on [a, b]. Prove that
Suppose that a < b. Decide which of the following statements are true and which are false. Prove the true ones and give counterexamples for the false ones. a) If f is bounded on [a, b], if g is absolutely integrable on (a, b), and if |(x)| < g(x) for all x ∈ (a, b), then f is absolutely
Evaluate the following improper integrals.a)(b)(c)d)
For each of the following, find all values of p ∈ R for which f is improperly integrable on I. a) f(x) = 1/xp I = (l, ∞) b) f(x) = l/xp, I = (0, 1) c) f(x) = 1/(x logp x), I = (e, ∞) d) f(x) = 1/(1/ xp), I = (0, ∞) e) f(x) = loga x/xp, where a > 0 is fixed, and I = (1, ∞)
Let p > 0. Show that x/xp is improperly integrable on [1, ∞) and that cos x/logp x is improperly integrable on [e, ∞).
Decide which of the following functions are improperly integrable on I. a) f(x) = sin x, I = (0, ∞) b) f(x)= l/x2, I = [-1. 1] c) f(x) = x-l sin(x-1), I = (l, ∞) d) f(x) = log(sin x), I = (0, 1) e) f(x) = (1 - cos x)/x2, I = (0, ∞)
Suppose that f, g are nonnegative and locally integrable on [a, b) and thatexists as an extended real number.a) Show that if 0 < L < ∞ and g is improperly integrable on [a, b), then so is f.b) Show that if 0 < L < ∞ and g is not improperly integrable on [a, b), then neither is
a) Suppose that f is improperly integrable on [0, ˆž). Prove that if L = limx†’ˆž f(x) exists, then L = 0.b) LetProve that f is improperly integrable on (0, ˆž) but limx†’ˆž, does not exist.
Prove that if f is absolutely integrable on (l, ˆž), then
Assuming e = limn†’ˆž, ˆ‘nk=0 1/k! (see Example 7.45), prove that
a) Show that 4k/(4k2 - 1) > 1/k for k ∈ N.b) Prove thatfor all n ∈ N.c) Prove thatis not of bounded variation on [0,1].
a) Show that (8k2 + 2)/(4k2 - l)2 b) Prove thatfor n ˆˆ N.c) Prove thatis of bounded variation on [0,1].
Suppose that Ï• and Ïˆ are of bounded variation on a closed interval [a, b).a) Prove that Î±Ï• is of bounded variation on [a, b] for every a ˆˆ R.b) Prove that Ï•Ïˆ is of bounded variation on [a, b].c) If there is an Îµ0 > 0 such thatprove that 1/Ï• is of bounded variation
Suppose that ϕ is of bounded variation on a closed, bounded interval [a, b]. Prove that ϕ is continuous on (a, b) if and only if ϕ is uniformly continuous on {a, b).
a) If ϕ is continuous on a closed nondegenerate interval [a, b], differentiate on (a,b), and if ϕ' is bounded on (a, b), prove that ϕ is of bounded variation on [a, b]. b) Show that ϕ(x) = 3√x is of bounded variation on [0, 1] but ϕ' is unbounded on (0, 1).
Let P be a polynomial of degree N. a) Show that P is of bounded variation on any closed interval [a, b]. b) Obtain an estimate for Var(P) on [a, b], using values of the derivative P'(x) at no more than TV points.
Let ϕ be a function of bounded variation on [a, b] and Φ be its total variation function. Prove that if Φ is continuous at some point x0 ∈ (a, b), then ϕ is continuous at x0.
If f is integrable on [a, b], prove thatis of bounded variation on [a, b].
Suppose that f' exists and is integrable on [a ,b]. Prove that f is of bounded variation andIf f' is bounded rather than integrable, how do the upper and lower integrals of f' compare to the variation of f?
Suppose that f, g are convex on an interval I. Prove that f + g and cf are convex on I for any c > 0.
Suppose that fn is a sequence of functions convex on an interval I and thatexists for each x ˆˆ I. Prove that f is convex on I.
Prove that a function f is both convex and concave on f if and only if there exist m, b ∈ R such that f(x) = mx + b for x ∈ I.
Prove that f(x) = xp is convex on [0, ∞) for p > 1, and concave on [0, ∞) for 0 < p < 1.
Show that if f is increasing on [a, b], thenis convex on [a, b]. (Recall that by Exercise 5.1.8, f is integrable on [a, b].)
If f : [a, b] †’ R is integrable on[a, b], prove that

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