- 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
a + (b + c) = (a + b) + c, a(bc) = (ab)c.
Characteristic polynomial of a square matrix A
det(xIn - A), where A is an n x n matrix
An expression a + bi, where a (the real part) and b (the imaginary part) are real numbers
Degree of a polynomial (function)
The largest exponent on the variable in any of the terms of the polynomial (function)
p = g(x), where x represents demand and p represents price
Identity involving a trigonometric function of u/2.
Head minus tail (HMT) rule
An arrow with initial point (x1, y1 ) and terminal point (x2, y2) represents the vector <8x 2 - x 1, y2 - y19>
A shift of a graph to the left or right.
The numbers . . ., -3, -2, -1, 0,1,2,...2
Inverse relation (of the relation R)
A relation that consists of all ordered pairs b, a for which a, b belongs to R.
Measure of center
A measure of the typical, middle, or average value for a data set
Multiplicative inverse of a real number
The reciprocal of b, or 1/b, b Z 0
NINT (ƒ(x), x, a, b)
A calculator approximation to ?ab ƒ(x)dx
See Linear programming problem.
See Periodic function.
Zeros of a function that are real numbers.
The principle of experimental design that minimizes the effects of chance variation by repeating the experiment multiple times.
An end behavior asymptote that is a slant line
Sum of a finite arithmetic series
Sn = na a1 + a2 2 b = n 2 32a1 + 1n - 12d4,
See Power function.
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