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Questions and Answers of Calculus

Let a
Let Î´Ê¹ and Î´Ê¹Ê¹ be as in the preceding exercise and let Î´* be defined by
Let d be a gauge on I: = (a, b) and suppose that I does not have a d-fine partition.(a) Let c : = (a + b). Show that at least one of the intervals [a, c] and [c, b] does not have a Î´-fine
Let I := (a, b) and let f : I → R be a (not necessarily continuous) function. We say that f is ''locally bounded'' at c ∈ I if there exists δ(c) > 0 such that f is bounded on I ∩(c - δ(c), c
Let I := [a, b] and let f : I → R be continuous on I. If f has an absolute maximum [respectively, minimum] at an interior point c of I, show that f is not injective on I
Let f : |0, 1 → R be a continuous function that does not take on any of its values twice and with f(0) < f(1). Show that f is strictly increasing on [0, 1].
Let h : |0, 1| → R be a function that takes on each of its values exactly twice. Show that h cannot be continuous at every point. [If c1 < c2 are the points where h attains its supremum, show that
Let x ∈ R, x > 0. Show that if m, p ∈ Z, n, q ∈ N, and mq = np, then (x1/n)m = (x1/q)p
If x ∈ R, x > 0, and if r, s ∈ Q, show that xrxs = xrþs = xsxr and (xr)s = xrs = (xs)r.
Show that if I := [a, b] and f : I → R is increasing on I, then f is continuous at a if and only if f (a) = inf{ f(x) : x ∈ (a, b]}.
Let I ⊂ R be an interval and let f : I → R be increasing on I. Suppose that c ∈ I is not an endpoint of I. Show that f is continuous at c if and only if there exists a sequence (xn) in I such
Let I ⊂ R be an interval and let f : I → R be increasing on I. If c is not an endpoint of I, show that the jump jf (c) of f at c is given by inf{f(y) - f(x) : x < c < y, x, y ∈ I}
Let I := [0, 1] and let f : I → R be defined by f (x) := x for x rational, and f(x) := 1 - x for x irrational. Show that f is injective on I and that f (f(x)) = x for all x ∈ I. (Hence f is its
Use the definition to find the derivative of each of the following functions: (a) f(x) := x3 for x ∈ R. (b) g(x) := 1/x for x ∈ R; x ≠ 0. (c) h(x) = √x for x > 0. (d) k(x) := 1/√x for x > 0.
Let g : R → R be defined by g(x) := x2sin (1/x2) for x ≠ 0, and g(0) := 0. Show that g is differentiable for all x ∈ R. Also show that the derivative gʹ is not bounded on the interval (-1, 1).
Assume that there exists a function L : (0; ∞) → R such that Lʹ (x) = 1/x for x > 0. Calculate the derivatives of the following functions: (a) f(x) := L(2x + 3) for x > 0. (b) g(x) := (L(x2))3
Let f : I → R be differentiable at c ∈ I. Establish the Straddle Lemma: Given ε > 0 there exists δ(e) > 0 such that if u, v ∈ I satisfy c - δ(e) < u < c < v < c + δ(e), then we have f(v) -
Prove Theorem 6.1.3(a), (b).
Differentiate and simplify: (a) f(x) := x/1 + x2, (b) g(x) := √5 - 2x + x2, (c) h(x) := (sin xk)m for m, k ∈ N. (d) k(x) := tan (x2) for |x| < √π/2.
Suppose that f : R → R is differentiable at c and that f(c) = 0. Show that g(x) := |f(x)| is differentiable at c if and only if fʹ(c) = 0.
Determine where each of the following functions from R to R is differentiable and find the derivative: (a) f(x) := |x| + |x + 1|. (b) g(x) : = 2x + |x|. (c) h(x) := x|x|. (d) k(x) := |sin x|.
For each of the following functions on R to R, find points of relative extrema, the intervals on which the function is increasing, and those on which it is decreasing: (a) f(x) := x2 - 3x + 5. (b)
Let g : R → R be defined by g(x) := x + 2x2 sin(1/x) for x ≠ 0 and g(0) := 0. Show that gʹ(0) = 1, but in every neighborhood of 0 the derivative gʹ(x) takes on both positive and negative
If h(x) := 0 for x < 0 and h(x) := 1 for x > 0, prove there does not exist a function f : R → R such that fʹ(x) = h(x) for all x ∈ R. Give examples of two functions, not differing by a constant,
Let f : (0, ˆž) †’ R be differentiable on (0, 1) and assume that fÊ¹(x) †’ b as x †’ 1.(b) Show that if f(x) †’ a as x †’ ˆž, then b = 0.
Let I := (a, b) and let f : I †’ R be differentiable at c ˆˆ I. Show that for every Îµ > 0 there exists Î´ > 0 such that if 0
A differentiable function f : I †’ R is said to be uniformly differentiable on I := (a, b) if for every Îµ > 0 there exists Î´ > 0 such that if 0Show that if f is
Find the points of relative extrema, the intervals on which the following functions are increasing, and those on which they are decreasing: (a) f(x) := x + 1/x for x ≠ 0. (b) g(x) := x/(x2 + 1) for
Find the points of relative extrema of the following functions on the specified domain: (a) f(x) := x2 - 1 for - 4 < x < 4. (b) g(x) := 1 - (x - 1)2/3 for 0 < x < 2; (c) h(x) := x|x2 - 12| for - 2 <
Let f : (a, b) †’ R be continuous on [a, b] and differentiable in (a, b). Show that if then fÊ¹(a) exists and equals A. [Use the definition of fÊ¹(a) and the Mean Value Theorem.]
Let f : R → R be defined by f(x) := 2x4 + x4 sin(1/x) for x ≠ 0 and f(0) := 0. Show that f has an absolute minimum at x ¼ 0, but that its derivative has both positive and negative values in
Evaluate the following limits:(a)(b)(c)(d)
Evaluate the following limits:(a)(b) (c) (d)
Let f be differentiable on (0, 1) and suppose that Show that [f(x) = exf(x)/ex.]
In addition to the suppositions of the preceding exercise, let g(x) > 0 for x ˆˆ (a, b), x ‰ c. If A > 0 and B = 0, prove that we must have If A
Let f(x) := x2sin (1/x) for 0 0 for x ‰ 0. Show that does not exist.
Evaluate the following limits.(a)(b)
Evaluate the following limits, where the domain of the quotient is as indicated.(a)(b) (c) (d)
Evaluate the following limits:(a)(b) (c) (d)
Evaluate the following limits:(a)(b) (c) (d)
Let h(x) := e-1/x2 for x ‰ 0 and h(0) := 0. Show that h(n)(0) = 0 for all n ˆˆ N. Conclude that the remainder term in Taylor's Theorem for x0 = 0 does not converge to zero as n
Calculate e correct to seven decimal places.
Determine whether or not x = 0 is a point of relative extrema of the following functions: (a) f(x) := x3 + 2. (b) g(x) := sin x - x. (c) h(x) := sin x + 1/6 x3. (d) k(x) := cos x - 1 + 1/2 x2:
Let f be continuous on [a, b] and assume the second derivative fʹʹ exists on (a, b). Suppose that the graph of f and the line segment joining the points (a, f(a)) and (b, f(b)) intersect at a point
If I := (0, 4), calculate the norms of the following partitions: (a) P1 := (0, 1, 2, 4). (b) P2 := (0, 2, 3, 4). (c) P3 := (0, 1, 1.5, 2, 3.4, 4). (d) P4 := (0, .5, 2.5, 3.5, 4)
Consider the Dirichlet function, introduced in Example 5.1.6(g), defined by f(x) := 1 for x ∈(0, 1) rational and f(x) := 0 for x ∈ (0, 1) irrational. Use the preceding exercise to show that f is
Suppose that c 0 for x ˆˆ [c, d] and Ï†(x) = 0 elsewhere in [a, b], prove that Ï† ˆˆ R[a, b] and that ˆ«Rb Ï† = a(d - c). [Given Îµ > 0 let Î´Îµ := Îµ/4a and show that if
Let 0(a) Show that qi satisfies 0 (b) Show that Q(qi) (xi - xi-1) = 1/3 (x3i - x3i-1.(d) Use the argument in Example 7.1.4(c) to show that Q ˆˆ R[a, b] and
If f ∈ R[a, b] and c ∈ R, we define g on [a + c, b + c] by g(y) := f(y - c). Prove that g ∈ R[a + c, b + c] and that ∫b+ca+c g = ∫ba f . The function g is called the c-translate of f.
If f (x) := x2 for x ˆˆ (0, 4), calculate the following Riemann sums, where has the same partition points as in Exercise 1, and the tags are selected as indicated.(a) with the tags at the
Show that f : [a, b] †’ R is Riemann integrable on [a, b] if and only if there exists L ˆˆ R such that for every Îµ > 0 there exists Î´Îµ > 0 such that if is any tagged partition
Let :{(Ii , ti)}ni=1 be a tagged partition of [a, b] and let c1 (a) If u belongs to a subinterval Ii whose tag satisfies c1 .(b) If v ˆˆ [a, b] and satisfies c1 + , then the tag ti of any
(a) Let f(x) := 2 if 0 < x < 1 and f(x) := 1 if 1 < x < 2. Show that f ∈ R[0, 2] and evaluate its integral.(b) Let h(x) := ∈ if 0 < x < 1, h(1) := 3 and h(x) := 1 if 1 < x
If f ˆˆ R[a, b] and if is any sequence of tagged partitions of [a, b] such that prove that
Let f : [a, b] †’ R. Show that f ˆ‰ R[a, b] if and only if there exists Îµ0 > 0 such that for every n ˆˆ N there exist tagged partitions with such that
If f and g are continuous on [a, b] and if ∫ba f = ∫ba g, prove that there exists c ∈ [a, b] such that f(c) = g(c).
If f is bounded by M on [a, b] and if the restriction of f to every interval [c, b] where c ∈ [a, b] is Riemann integrable, show that f ∈ R[a, b] and that ∫bc f → ∫ba f as c → a+. [Let
If f is bounded and there is a finite set E such that f is continuous at every point of [a, b]\E, show that f ∈ R[a, b].
If f is continuous on [a, b], a < b, show that there exists c ∈ [a, b] such that we have ∫ba f = f(c)(b - a). This result is sometimes called the Mean Value Theorem for Integrals.
If f and g are continuous on [a, b] and g(x) > 0 for all x ∈ [a, b], show that there exists c ∈ [a, b] such that ∫ba fg = f(c) ∫ba g. Show that this conclusion fails if we do not have g(x) >
Let f be continuous on [a, b], let f(x) > 0 for x ∈ [a, b], and let Mn :(∫ba fn)1/n. Show that lim (Mn) = sup{f(x) : x ∈ [a, b]g.
Suppose that a > 0 and that f ∈ R[-a, a].(a) If f is even (that is, if f (-x) = f(x) for all x ∈ [0, a], show that ∫a-a f = 2 ∫a0 f.(b) If f is odd (that is, if f (-x) = -f(x) for all x
Suppose that f is continuous on [a, b], that f(x) > 0 for all x ∈ [a, b] and that ∫ba f = 0. Prove that f(x) = 0 for all x ∈ [a, b].
Extend the proof of the Fundamental Theorem 7.3.1 to the case of an arbitrary finite set E.
Let f : [a, b] → R be continuous on [a, b] and let v : [c, d] → R be differentiable on [c, d] with v ([c, d)] ⊂ [a, b]. If we define G(x) := ∫v(x)a f , show that Gʹx = f(v(x)) ∙ vʹ(x) for
Let f : [0, 3] → R be defined by f(x) := x for 0 < x < 1, f(x) := 1 for 1 < x < 2 and f(x) := x for 2 < x < 3. Obtain formulas for F(x) := ∫x0 f and sketch the graphs of f and F. Where is F
The function g is defined on [0, 3] by g(x) := -1 if 0 < x < 2 and g(x) := 1 if 2 < x < 3. Find the indefinite integral G(x) = ∫x0 g for 0 < x < 3, and sketch the graphs of g and G. Does Gʹ (x) =
Show there does not exist a continuously differentiable function f on [0, 2] such that f(0) = -1, f(2) = 4, and fʹ(x) < 2 for 0 < x < 2. (Apply the Fundamental Theorem.)
Explain why Theorem 7.3.8 and/or Exercise 7.3.17 cannot be applied to evaluate the following integrals, using the indicated substitution.(a)(b) (c) (d)
Let f ; g ∈ R[a, b].(a) If t ∈ R, show that ∫ba (tf ± g)2 > 0.(b) Use (a) to show that 2|∫ba f g| < t ∫ba f2 + (1/t) ∫ba g2 for t > 0.(c) If ∫ba f2 = 0, show that ∫ba f g =
We have seen in Example 7.1.7 that Thomae's function is inR[0, 1] with integral equal to 0. Can the Fundamental Theorem 7.3.1 be used to obtain this conclusion? Explain your answer.
Let F(x) be defined for x > 0 by F(x) := (n - 1)x - (n - 1)n/2 for x ∈ [n - 1, n], n ∈ N. Show that F is continuous and evaluate Fʹ(x) at points where this derivative exists. Use this result
Let f (x) := |x| for -1 < x < 2. Calculate L(f; P) and U(f; P) for the following partitions: (a) P1 := (-1, 0, 1, 2). (b) P2 := (-1, -1/2, 0, 1/2, 1, 3/2, 2).
If f is a bounded function on [a, b] such that f(x) = 0 except for x in {c1, c2, . . . , cn} in [a, b], show that U(f) = L(f) = 0.
Let f(x) = x2 for 0 < x < 1. For the partition Pn := (0, 1/n, 2/n, . . . , (n - 1)/n, 1), calculate L(f, Pn) and U(f, Pn), and show that L(f) = U(f) = 1/3. (Use the formula 12 + 22 + ∙ ∙ ∙+m2 =
Let Pε be the partition whose existence is asserted in the Integrability Criterion 7.4.8. Show that if P is any refinement of Pε, then U(f; P) - L(f; P) < ε.
Write out the proofs that a function f on [a, b] is Darboux integrable if it is either(a) Continuous.(b) Monotone.
Let f be defined on I := (a, b) and assume that f satisfies the Lipschitz condition |f(x) - f(y)| < K|x - y| for all x, y in I. If Pn is the partition of I into n equal parts, show that 0 < U (f; Pn)
Prove if f(x) := c for x ∈ (a, b), then its Darboux integral is equal to c(b - a).
Let f and g be bounded functions on I := (a, b). If f(x) < g(x) for all x ∈ I, show that L(f) < L(g) and U(f) < U(g).
(a) Prove that if g(x) := 0 for 0 < x < 1/2 and g(x) := 1 for 1/2 < x < 1, then the Darboux integral of g on [0, 1] is equal to 1/2. (b) Does the conclusion hold if we change the value of g at the
Let f be continuous on I := (a, b) and assume f(x) > 0 for all x ∈ I. Prove if L(f) = 0, then f(x) = 0 for all x ∈ I.
Let f1 and f2 be bounded functions on (a, b). Show that L(f1) + L(f2) < L(f1 + f2).
Note that ∫10 (1 - x2)1/2 dx = p/4. Explain why the error estimates given by formulas (4), (7), and (10) cannot be used. Show that if h(x) = (1 - x2)1/2 for x in [0, 1], then Tn(h) < π/4 < Mn(h).
Show that if fʹʹ(x) > 0 on (a, b) (that is, if f is convex on [a, b]), then for any natural numbers m, n we have Mn(f) < ∫ba f(x)dx < Tm(f): If fʹʹ(x) < 0 on [a, b], this inequality is reversed.
Show that lim((cos πx)2n) exists for all x ∈ R. What is its limit?
Show that if a > 0, then the convergence of the sequence in Exercise 6 is uniform on the interval [a, ∞), but is not uniform on the interval (0, ∞).
Show that if a > 0, then the sequence (n2x2e-nx) converges uniformly on the interval [a, ∞), but that it does not converge uniformly on the interval [0, ∞).
Show that if (fn), (gn) converge uniformly on the set A to f, g, respectively, then (fn + gn) converges uniformly on A to f + g.
Show that if fn(x) := x + 1/n and f(x) := x for x ∈ R, then (fn) converges uniformly on R to f, but the sequence (f2n) does not converge uniformly on R. (Thus the product of uniformly convergent
Let (fn), (gn) be sequences of bounded functions on A that converge uniformly on A to f, g, respectively. Show that (fngn) converges uniformly on A to fg.
Let (fn) be a sequence of functions that converges uniformly to f on A and that satisfies |fn (x)| < M for all n ∈ N and all x ∈ A. If g is continuous on the interval = (-M; M), show that the
Evaluate lim(xn/(1 + xn)) for x ∈ R, x > 0.
Show that lim(Arctan nx) = (π/2)sgn x for x ∈ R.
Show that lim x2e-nx) = 0 and that lim (n2x2e-nx) = 0 for x ∈ R; x > 0.
Show that the sequence (xn/(1 + xn)) does not converge uniformly on [0, 2] by showing that the limit function is not continuous on [0, 2].
Let gn(x) := e-nx/n for x > 0, n ∈ N. Examine the relation between lim(gn) and lim(gʹn).
If a > 0, show that lim ∫πa (sin nx)/(nx)dx = 0. What happens if a = 0?
Let fn(x) := nx/(1 + nx) for x ∈ (0, 1). Show that (fn) converges non uniformly to an integrable function f and that ∫10 f(x)dx = lim ∫10 fn(x)dx.
Let gn(x) := nx(1 - x)n for x ∈ (0, 1), n ∈ N. Discuss the convergence of (gn) and (∫10 gndx).

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