Textbook Question
In Exercises 1–24, find the derivative of y with respect to the appropriate variable.
5. y = ln(sin²θ)
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.P.112a
112. True, or false? Give reasons for your answers.
a. 1/x⁴ = O(1/x² + 1/x⁴)
Verified step by step guidance1
Recall the definition of Big-O notation: A function \(f(x)\) is \(O(g(x))\) as \(x \to \infty\) if there exist positive constants \(C\) and \(M\) such that for all \(x > M\), \(|f(x)| \leq C |g(x)|\).
Identify the functions involved: \(f(x) = \frac{1}{x^4}\) and \(g(x) = \frac{1}{x^2} + \frac{1}{x^4}\).
Analyze the behavior of \(g(x)\) as \(x \to \infty\): Since \(\frac{1}{x^2}\) dominates \(\frac{1}{x^4}\) for large \(x\), \(g(x)\) behaves roughly like \(\frac{1}{x^2}\).
Compare \(f(x)\) and \(g(x)\): For large \(x\), \(\frac{1}{x^4}\) is smaller than or equal to a constant multiple of \(\frac{1}{x^2}\), so \(f(x)\) is bounded above by \(g(x)\) times some constant.
Conclude whether \(f(x) = O(g(x))\) holds by verifying if the inequality \(|f(x)| \leq C |g(x)|\) can be satisfied for some constants \(C\) and \(M\).
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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, showing how it behaves as the input approaches a limit, often infinity. It helps compare functions by their dominant terms, ignoring lower-order terms and constants.
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Sigma Notation
Asymptotic Comparison of Functions
Asymptotic comparison involves analyzing how two functions behave relative to each other as the variable grows large or approaches zero. It determines if one function grows faster, slower, or at the same rate as another.
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5:50Asymptotes of Hyperbolas
Properties of Rational Functions
Rational functions are ratios of polynomials. Understanding their behavior, especially for terms like 1/x² and 1/x⁴, is crucial to analyze limits and growth rates, as higher powers in the denominator lead to faster decay as x increases.
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06:21Properties of Functions
Related Practice
Textbook Question
112. True, or false? Give reasons for your answers.
c. ln x = o(x+1)
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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
In Exercises 25–30, use logarithmic differentiation to find the derivative of y with respect to the appropriate variable.
25. y = 2(x² + 1)/√(cos 2x)
Textbook Question
In Exercises 25–30, use logarithmic differentiation to find the derivative of y with respect to the appropriate variable.
27. y = (((t+1)(t-1))/((t-2)(t+3)))^5, t>2
Textbook Question
111. True, or false? Give reasons for your answers.
c. x = o(x + ln(x))