Textbook Question
Evaluate the integrals in Exercises 31–78.
49. ∫x3^(x²)dx
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 Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 7, Problem 7.PE.110f
110. Does f grow faster, slower, or at the same rate as g as x→∞? Give reasons for your answers.
f. f(x) = sech(x), g(x) = e^(-x)
Verified step by step guidance1
Recall the definitions of the functions: \(f(x) = \text{sech}(x)\) and \(g(x) = e^{-x}\). The hyperbolic secant function is defined as \(\text{sech}(x) = \frac{2}{e^{x} + e^{-x}}\).
Analyze the behavior of \(f(x)\) as \(x \to \infty\). Since \(e^{x}\) grows very large, the term \(e^{x} + e^{-x}\) is dominated by \(e^{x}\), so \(\text{sech}(x) \approx \frac{2}{e^{x}} = 2e^{-x}\) for large \(x\).
Compare this approximation of \(f(x)\) to \(g(x) = e^{-x}\). Notice that \(f(x)\) behaves like \(2e^{-x}\), which is just a constant multiple of \(g(x)\) as \(x \to \infty\).
Since \(f(x)\) and \(g(x)\) differ only by a constant factor in their dominant terms for large \(x\), they decay at the same exponential rate as \(x \to \infty\).
Conclude that \(f\) and \(g\) grow (or decay) at the same rate as \(x \to \infty\) because their leading terms have the same exponential order.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Asymptotic Behavior of Functions
Asymptotic behavior describes how functions behave as the input approaches a limit, often infinity. Understanding this helps compare growth rates by analyzing limits of ratios or differences as x→∞, revealing which function dominates or if they grow at similar rates.
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Asymptotes of Hyperbolas
Hyperbolic Secant Function (sech)
The hyperbolic secant, sech(x), is defined as 1/cosh(x), where cosh(x) = (e^x + e^{-x})/2. As x→∞, cosh(x) grows exponentially, so sech(x) approaches zero exponentially, behaving similarly to e^{-x} but with a specific constant factor.
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Exponential Decay and Comparison
Exponential decay functions like e^{-x} decrease rapidly as x→∞. Comparing f(x) = sech(x) to g(x) = e^{-x} involves examining their rates of decay by considering limits of their ratio, which determines if one decays faster, slower, or at the same rate.
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Related Practice
Textbook Question
133. What is the age of a sample of charcoal in which 90% of the carbon-14 originally present has decayed?
Textbook Question
110. Does f grow faster, slower, or at the same rate as g as x→∞? Give reasons for your answers.
c. f(x) = 10x^3 + 2x^2, g(x) = e^x
Textbook Question
In Exercises 1–24, find the derivative of y with respect to the appropriate variable.
11. y = 5x^(3.6)
Textbook Question
In Exercises 79–84, solve for y.
79. 3^y = 2^(y+1)
Textbook Question
113. The function f(x) = e^x + x, being differentiable and one-to-one, has a differentiable inverse f^(-1)(x). Find the value of df^(-1)/dx at the point f (ln 2).