Textbook Question
9. True, or false? As x→∞,
e. e^x = o(e^(2x))
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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 7.8.4e
4. Which of the following functions grow faster than x² as x→∞? Which grow at the same rate as x²? Which grow slower?
e. x^3 - x^2
Verified step by step guidance1
Identify the dominant term in the function \(x^3 - x^2\) as \(x \to \infty\). The dominant term is the one with the highest power of \(x\), which is \(x^3\) in this case.
Recall that the growth rate of a function as \(x \to \infty\) is determined by its highest power term. Here, \(x^3\) grows faster than \(x^2\) because the exponent 3 is greater than 2.
Compare the dominant term \(x^3\) with \(x^2\): since \(x^3\) grows faster than \(x^2\), the function \(x^3 - x^2\) grows faster than \(x^2\) as \(x \to \infty\).
To confirm, consider the limit \(\lim_{x \to \infty} \frac{x^3 - x^2}{x^2} = \lim_{x \to \infty} (x - 1) = \infty\), which shows the function grows faster than \(x^2\).
Therefore, \(x^3 - x^2\) grows faster than \(x^2\) as \(x\) approaches infinity.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Asymptotic Growth Rates
Asymptotic growth rates describe how functions behave as the input variable approaches infinity. Comparing growth rates helps determine which functions increase faster, slower, or at the same rate by focusing on dominant terms and ignoring lower-order terms.
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Intro To Related Rates
Dominant Term in Polynomials
In polynomial functions, the term with the highest exponent dominates the function's behavior for large values of x. For example, in x³ - x², the x³ term grows faster than x², so the overall function behaves like x³ as x approaches infinity.
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07:00Taylor Polynomials
Big-O and Big-Theta Notation
Big-O and Big-Theta notations classify functions by their growth rates. Big-O gives an upper bound, while Big-Theta indicates tight bounds, meaning two functions grow at the same rate. These notations help compare functions like x² and x³ in terms of their growth as x→∞.
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2:02Simplifying Trig Expressions Example 1
Related Practice
Textbook Question
2. Which of the following functions grow faster than e^x as x→∞? Which grow at the same rate as e^x? Which grow slower?
e. e^(-x)
Textbook Question
112. True, or false? Give reasons for your answers.
e. sec^(-1)x = O(1)
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:
e. Plot the functions f and g, the identity, the two tangent lines, and the line segment joining the points (x_0, f(x_0)) and (f(x_0), x_0). Discuss the symmetries you see across the main diagonal y=x.
72. y= 2-x-x³, -2 ≤ x ≤ 2, x_0 = 3/2
Textbook Question
1. Which of the following functions grow faster than e^x as x→∞? Which grow at the same rate as e^x? Which grow slower?
e. (3/2)^x
Textbook Question
3. Which of the following functions grow faster than x² as x→∞? Which grow at the same rate as x²? Which grow slower?
e. x ln(x)