Ch. 7 - Transcendental Functions

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 7 - Transcendental Functions

Problem 7.112e

Chapter 7, Problem 7.112e

112. True, or false? Give reasons for your answers.
e. sec^(-1)x = O(1)

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Understand the notation: \(\sec^{-1} x\) typically denotes the inverse secant function, also written as \(\operatorname{arcsec}(x)\), which gives the angle whose secant is \(x\).

Recall the definition of Big O notation: \(f(x) = O(g(x))\) means there exist constants \(C > 0\) and \(x_0\) such that for all \(x > x_0\), \(|f(x)| \leq C |g(x)|\). Here, the claim is \(\sec^{-1} x = O(1)\), meaning \(\sec^{-1} x\) is bounded by a constant as \(x\) grows large.

Analyze the behavior of \(\sec^{-1} x\) as \(x \to \infty\): Since \(\sec \theta = x\), for large \(x\), \(\theta = \sec^{-1} x\) approaches \(\frac{\pi}{2}\) from above or below, so \(\sec^{-1} x\) approaches a finite limit.

Conclude that since \(\sec^{-1} x\) approaches a finite constant as \(x\) grows large, it is bounded and thus \(\sec^{-1} x = O(1)\) is true.

Summarize: The statement is true because the inverse secant function does not grow without bound; it approaches a finite limit, satisfying the Big O condition with a constant function.

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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, O(1) means the function remains bounded by a constant regardless of input size.

Behavior of the Inverse Secant Function (sec⁻¹x)

The inverse secant function, sec⁻¹x, is defined for |x| ≥ 1 and returns the angle whose secant is x. As x grows large, sec⁻¹x increases without bound, meaning it does not remain constant or bounded. Understanding its domain and range is crucial for analyzing its growth.

Asymptotic Analysis of Trigonometric Inverse Functions

Analyzing the asymptotic behavior of inverse trigonometric functions involves studying their limits as the input approaches infinity or other critical points. Unlike bounded functions like arcsin or arctan, sec⁻¹x grows logarithmically or unbounded, affecting its classification in Big O terms.

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Textbook Question

In Exercises 67–72, you will explore some functions and their inverses together with their derivatives and tangent line approximations at specified points. Perform the following steps using your CAS:

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