 10.2.1: Show that Hake's Theorem 10.2.1 can be given the following sequenti...
 10.2.2: (a) Apply Hake'sTheorem to conclude that g(x) := l/x2/3forx E (O,I]...
 10.2.3: Apply Hake's Theorem to g(x) := (1  x)1/2 for x E [0. 1) and g(1)...
 10.2.4: Suppose that 1 E n*[a,c]forallc E (a, b) andthatthereexistsy E (a,b...
 10.2.5: Show that the function gl(x) :=xI/2sin(1/x) for x E (O,I]andgl(O) ...
 10.2.6: Show that the following functions (properly defined when necessary)...
 10.2.7: Determine whether the following integrals are convergentor divergen...
 10.2.8: If 1 E n[a, b], showthat1 E [a, b].
 10.2.9: If 1 E [a, b], showthat 12 is notnecessarilyin [a, b].
 10.2.10: If I, g E [a, b] and if g is bounded and monotone, show that Ig E [...
 10.2.11: (a) Give an example of a function I E R*[O, 1] such that max{f,O} d...
 10.2.12: Write out the details of the proofthat min{f, g} E R*[a, b] in Theo...
 10.2.13: Write out the details of the proofs of Theorem 10.2.11.
 10.2.14: Givean I E [a, b] with I not identically0, but suchthat 11/11 =O.
 10.2.15: If I, g E [a, b], show that 111/11  IIglll~ 1I/:l: gll.
 10.2.16: Establish the easy part of the Completeness Theorem 10.2.12.
 10.2.17: If In(x) := xn for n E N, show that In E [0, 1] and that IIIn II ~ ...
 10.2.18: Let gn(x) := 1 for x E [1, I/n), let gn(x) := nx for x E [I/n, ...
 10.2.19: Lethn(x) :=nforx E (0, I/n)andhn(x) :=Oelsewherein[O, 1]. Does ther...
 10.2.20: Let kn(x) := n for x E (0, II n2) and kn(x) := 0 elsewhere in [0, 1...
Solutions for Chapter 10.2: Improper and Lebesgue Integrals
Full solutions for Introduction to Real Analysis  3rd Edition
ISBN: 9780471321484
Solutions for Chapter 10.2: Improper and Lebesgue Integrals
Get Full SolutionsSince 20 problems in chapter 10.2: Improper and Lebesgue Integrals have been answered, more than 4984 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Introduction to Real Analysis, edition: 3. This expansive textbook survival guide covers the following chapters and their solutions. Introduction to Real Analysis was written by and is associated to the ISBN: 9780471321484. Chapter 10.2: Improper and Lebesgue Integrals includes 20 full stepbystep solutions.

Basic logistic function
The function ƒ(x) = 1 / 1 + ex

Bounded interval
An interval that has finite length (does not extend to ? or ?)

Complex fraction
See Compound fraction.

Compounded k times per year
Interest compounded using the formula A = Pa1 + rkbkt where k = 1 is compounded annually, k = 4 is compounded quarterly k = 12 is compounded monthly, etc.

Cone
See Right circular cone.

Cycloid
The graph of the parametric equations

First octant
The points (x, y, z) in space with x > 0 y > 0, and z > 0.

Inequality
A statement that compares two quantities using an inequality symbol

Interquartile range
The difference between the third quartile and the first quartile.

Limaçon
A graph of a polar equation r = a b sin u or r = a b cos u with a > 0 b > 0

Lower bound for real zeros
A number c is a lower bound for the set of real zeros of ƒ if ƒ(x) Z 0 whenever x < c

Negative association
A relationship between two variables in which higher values of one variable are generally associated with lower values of the other variable.

Normal distribution
A distribution of data shaped like the normal curve.

Right angle
A 90° angle.

Scalar
A real number.

Standard representation of a vector
A representative arrow with its initial point at the origin

Standard unit vectors
In the plane i = <1, 0> and j = <0,1>; in space i = <1,0,0>, j = <0,1,0> k = <0,0,1>

Symmetric about the yaxis
A graph in which (x, y) is on the graph whenever (x, y) is; or a graph in which (r, ?) or (r, ?, ?) is on the graph whenever (r, ?) is

Vertex of a cone
See Right circular cone.

yzplane
The points (0, y, z) in Cartesian space.