Textbook Question
In Exercises 1–24, find the derivative of y with respect to the appropriate variable.
13. y = (x+2)^(x+2)
Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
Not the one you use?Change textbookSelect a different textbook
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.P.111e
111. True, or false? Give reasons for your answers.
e. arctan x = O(1)
Verified step by step guidance1
Recall the definition of Big O notation: a function \(f(x)\) is \(O(1)\) as \(x \to \infty\) if there exists a constant \(M\) such that \(|f(x)| \leq M\) for all sufficiently large \(x\).
Consider the behavior of \(\arctan x\) as \(x \to \infty\). The arctangent function approaches a finite limit, specifically \(\frac{\pi}{2}\).
Since \(\arctan x\) approaches a finite constant, it is bounded for all \(x\), meaning there exists some constant \(M\) such that \(|\arctan x| \leq M\) for all \(x\).
Therefore, by the definition of Big O notation, \(\arctan x\) is indeed \(O(1)\) as \(x \to \infty\) because it does not grow without bound but stays within a fixed range.
Conclude that the statement '\(\arctan x = O(1)\)' is true, and the reasoning is based on the boundedness and finite limit of the arctangent function.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
1mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Big O Notation
Big O notation describes the upper bound of a function's growth rate as the input approaches a limit, often infinity. It characterizes how a function behaves asymptotically, ignoring constant factors and lower-order terms. For example, f(x) = O(1) means the function is bounded by a constant as x grows large.
Recommended video:
04:22
Sigma Notation
Behavior of arctan(x) as x → ∞
The arctan function approaches a finite limit, π/2, as x approaches infinity. This means arctan(x) is bounded and does not grow without bound, which is crucial when analyzing its asymptotic behavior or classifying it using Big O notation.
Recommended video:
03:39Integrals of Natural Exponential Functions (e^x)
Asymptotic Boundedness
A function is asymptotically bounded if it remains within fixed limits as the input grows large. Since arctan(x) approaches π/2, it is bounded above and below, making it a constant order function, which aligns with the definition of O(1).
Recommended video:
5:50Asymptotes of Hyperbolas
Related Practice
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.
a. Find the volume of the solid.
Textbook Question
In Exercises 25–30, use logarithmic differentiation to find the derivative of y with respect to the appropriate variable.
29. y = (sin θ)^√θ
Textbook Question
118. A particle is traveling upward and to the right along the curve y=ln(x). Its x-coordinate is increasing at the rate (dx/dt)=√x m/sec. At what rate is the y-coordinate changing at the point (e², 2)?
Textbook Question
7. What integrals lead to logarithms? Give examples. What are the integrals of tan x, cot x, sec x, and csc x?
Textbook Question
19. Center of mass Find the center of mass of a thin plate of constant density covering the region in the first and fourth quadrants enclosed by the curves y=1/(1+x²) and y=-1/(1+x²) and by the lines x=0 and x=1.