Textbook Question
13. For what x>0 does x^(x^x) = (x^x)^x? Give reasons for your answer.
Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 7, Problem 7.AAE.17
17. Even-odd decompositions
b. If f(x) = f_E(x) + f_O(x) is the sum of an even function f_E(x) and an odd function f_O(x), then show that
f_E(x) = (f(x)+f(-x))/2 and f_O(x) = (f(x)-f(-x))/2
Verified step by step guidance1
Recall the definitions of even and odd functions: an even function satisfies \(f_E(x) = f_E(-x)\), and an odd function satisfies \(f_O(x) = -f_O(-x)\).
Given that \(f(x) = f_E(x) + f_O(x)\), write the expression for \(f(-x)\) by substituting \(-x\) into the function: \(f(-x) = f_E(-x) + f_O(-x)\).
Use the properties of even and odd functions to rewrite \(f(-x)\) as \(f(-x) = f_E(x) - f_O(x)\), since \(f_E(-x) = f_E(x)\) and \(f_O(-x) = -f_O(x)\).
Add the two equations \(f(x) = f_E(x) + f_O(x)\) and \(f(-x) = f_E(x) - f_O(x)\) to isolate \(f_E(x)\): \(f(x) + f(-x) = 2 f_E(x)\).
Similarly, subtract \(f(-x)\) from \(f(x)\) to isolate \(f_O(x)\): \(f(x) - f(-x) = 2 f_O(x)\). Then solve for \(f_E(x)\) and \(f_O(x)\) to get \(f_E(x) = \frac{f(x) + f(-x)}{2}\) and \(f_O(x) = \frac{f(x) - f(-x)}{2}\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Even and Odd Functions
An even function satisfies f(x) = f(-x) for all x, meaning its graph is symmetric about the y-axis. An odd function satisfies f(-x) = -f(x), showing symmetry about the origin. Understanding these definitions is essential to decompose any function into even and odd parts.
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Properties of Functions
Function Decomposition
Any function f(x) can be expressed as the sum of an even function f_E(x) and an odd function f_O(x). This decomposition helps analyze the function's symmetry properties and simplifies integration and other operations by separating symmetric components.
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Algebraic Manipulation for Decomposition
To find f_E(x) and f_O(x), use the definitions of even and odd functions to form equations involving f(x) and f(-x). Adding and subtracting these equations isolates the even and odd parts, leading to the formulas f_E(x) = (f(x)+f(-x))/2 and f_O(x) = (f(x)-f(-x))/2.
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Related Practice
Textbook Question
Find the areas between the curves y=2(log_2(x))/x and y=2(log_4(x))/x and the x-axis from x=1 to x=e. What is the ratio of the larger area to the smaller?
Textbook Question
Find the limits in Exercises 1–6.
5. lim(n→∞) (1/(n+1) + 1/(n+2) + ... + 1/(2n))
Textbook Question
In Exercises 9 and 10, use implicit differentiation to find dy/dx.
9. y^e^x = x^y + 1
Textbook Question
7. Let A(t) be the area of the region in the first quadrant enclosed by the coordinate axes, the curve y=e^(-x), and the vertical line x=t, t>0. Let V(t) be the volume of the solid generated by revolving the region about the x-axis. Find the following limits.
a. lim(x→∞)A(t)
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Textbook Question
20. Solid of revolution The region between the curve y=1/(2√x) and the x-axis from x=1/4 to x=4 is revolved about the x-axis to generate a solid.
b. Find the centroid of the region.
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