- 8.3.1: Show that if x > 0 and ifn > 2x, thenI ( X xn )I2xn+1eX_ 1+-+...+- ...
- 8.3.2: Calculate e correct to 5 decimal places.
- 8.3.3: Show that if 0 ::: x ::: a and n EN, thenx xn X xn-l eaxn1+-+...+- ...
- 8.3.4: Showthatif n ~ 2, then( 1 1 ) e o < en! - 1+ 1+ - + .. . + - n! < -...
- 8.3.5: Ifx ~ 0 andn E N, showthat1-= 1 -x+x 2 -x 3 +...+(-x) n-l +-(._x)n ...
- 8.3.6: Usetheformula in theprecedingexerciseto calculateIn 1.1andIn 1.4acc...
- 8.3.7: Showthat In(e/2) = 1 -In 2. Use this result to calculate In 2 accur...
- 8.3.8: Let I : JR-+ JR be such that I' (x) = I (x) for all x e JR. Show th...
- 8.3.9: Let ak > 0 for k = 1, "', n and let A := (a1+... + an)/n be the ari...
- 8.3.10: Evaluate L'(I) by using the sequence (1 + l/n) and the fact that e ...
- 8.3.11: Estaglish the assertions in Theorem 8.3.11.
- 8.3.12: Establish the assertions in Theorem 8.3.12.
- 8.3.13: (a)Establish the assertions in Theorem 8.3.12.Show that if ex> 0, t...
- 8.3.14: Prove that if a > 0, a =F 1, then alogax= x for all x e (0,00) and ...
- 8.3.15: If a > 0, a =F 1, show that the function x 1-+logox is differentiab...
- 8.3.16: If a > 0, a =F1, and x and y belong to (0,00), prove that logo(xy) ...
- 8.3.17: If a > 0, a =F 1, and b > 0, b =F 1, show that(In b logox = In a)10...
Solutions for Chapter 8.3: The Exponential and Logarithmic Functions
Full solutions for Introduction to Real Analysis | 3rd Edition
Addition property of inequality
If u < v , then u + w < v + w
See Polar coordinates.
Direction angle of a vector
The angle that the vector makes with the positive x-axis
A function that can be written in the form ƒ(x) = mx + b, where and b are real numbers
A local maximum or a local minimum
Negative linear correlation
See Linear correlation.
See Linear programming problem.
Order of magnitude (of n)
See Parametric equations.
The numbers (r, ?) that determine a point’s location in a polar coordinate system. The number r is the directed distance and ? is the directed angle
A function of the form ƒ(x) = k . x a, where k and a are nonzero constants. k is the constant of variation and a is the power.
Reciprocal of a real number
See Multiplicative inverse of a real number.
Reflexive property of equality
a = a
See Elementary row operations.
A procedure for fitting a curve y = a sin (bx + c) + d to a set of data
A line that is neither horizontal nor vertical
Sum of functions
(ƒ + g)(x) = ƒ(x) + g(x)
Term of a polynomial (function)
An expression of the form anxn in a polynomial (function).
The square of the standard deviation.
Vertex of a cone
See Right circular cone.