- 7.2.1: Let f : [a, b] JR.Show that f ~ n[a, b] if and only if there exists...
- 7.2.2: Consider the function h defined by h(x) := x + 1 for x E [0, 1] rat...
- 7.2.3: Let H(x) := k for x = 1/ k (k e N) and H(x) := 0 elsewhere on [0,1]...
- 7.2.4: If a(x) := -x and w(x) := x and if a(x) ~ f(x) ~ w(x) for all x e [...
- 7.2.5: If J is any subinterval of [a, b] and if q1/x) := 1 for x e J and q...
- 7.2.6: If 1/1: [a, b] ~ JRtakes on only a finite number of distinct values...
- 7.2.7: If S(f; p) is any Riemann sum of f : [a, b] ~ JR, show that there e...
- 7.2.8: Suppose that f is continuous on [a, b], that f(x) ~ 0 for all x e [...
- 7.2.9: Show that the continuity hypothesis in the preceding exercise canno...
- 7.2.10: If f and g are continuous on [a, b] and if f:f = f:g, prove that th...
- 7.2.11: If f is boundedbyM on [a,b]andif therestrictionof f to everyinterva...
- 7.2.12: Show that g(x) := sin(l/x) for x e (0, 1] and g(O) := 0 belongs to ...
- 7.2.13: Givean exampleof a functionf : [a, b] ~ JRthat is in 'R.[c,b] for e...
- 7.2.14: Suppose that f : [a, b] ~ JR, that a = Co < CI < . .. < cm =b and t...
- 7.2.15: If f is bounded and there is a finite set E such that f is continuo...
- 7.2.16: If f is continuous on [a, b], a < b, show that there exists c E [a,...
- 7.2.17: If f and g are continuous on [a, b] and g(x) > 0 for all x e [a, b]...
- 7.2.18: Let f be continuous on [a, b], let f(x) ~ 0 for x e [a, b], and let...
- 7.2.19: Supposethata > 0 and that f e 'R.[-a, a].(a) If f is even(thatis, i...
- 7.2.20: Suppose that f : [a, b] ~ JRand that n eN. Let'P n be the partition...
- 7.2.21: If f is continuous on [-a, a], show that J~a f(x2) dx = 21;f(x2) dx.
- 7.2.22: Iff is continuous on [-1, 1], show that JOlr/2f(cosx)dx = J;/2f(sin...
Solutions for Chapter 7.2: Riemann Integrable Functions
Full solutions for Introduction to Real Analysis | 3rd Edition
Addition property of equality
If u = v and w = z , then u + w = v + z
The numbers in Pascal’s triangle: nCr = anrb = n!r!1n - r2!
A function is bounded above if there is a number B such that ƒ(x) ? B for all x in the domain of ƒ.
Two angles having the same initial side and the same terminal side
Facts collected for statistical purposes (singular form is datum)
The gathering and processing of numerical information
y = b.
The instantaneous rate of change of a position function with respect to time, p. 737.
A graph of a polar equation of the form r2 = a2 sin 2u or r 2 = a2 cos 2u.
The eight regions of space determined by the coordinate planes.
Quotient of functions
a ƒ g b(x) = ƒ(x) g(x) , g(x) ? 0
The portion of the domain applicable to the situation being modeled.
The distance from the center of an ellipse to a point on the ellipse along a line perpendicular to the major axis.
Shrink of factor c
A transformation of a graph obtained by multiplying all the x-coordinates (horizontal shrink) by the constant 1/c or all of the y-coordinates (vertical shrink) by the constant c, 0 < c < 1.
A measure of how a data set is spread
Standard form: equation of a circle
(x - h)2 + (y - k2) = r 2
Upper bound for real zeros
A number d is an upper bound for the set of real zeros of ƒ if ƒ(x) ? 0 whenever x > d.
A letter that represents an unspecified number.
Vertices of a hyperbola
The points where a hyperbola intersects the line containing its foci.
Zero factor property
If ab = 0 , then either a = 0 or b = 0.