Textbook Question
What can you conclude about the inverses of functions whose graphs are lines perpendicular to the line y=x?
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 7.8.4c
4. Which of the following functions grow faster than x² as x→∞? Which grow at the same rate as x²? Which grow slower?
c. x²e^(-x)
Verified step by step guidance1
Identify the given function: \(f(x) = x^{2} e^{-x}\).
Recall that \(e^{-x}\) can be rewritten as \(\frac{1}{e^{x}}\), which decreases very rapidly as \(x \to \infty\).
Compare the growth rates: \(x^{2}\) grows polynomially, while \(e^{-x}\) decays exponentially, so their product \(x^{2} e^{-x}\) tends to zero as \(x \to \infty\).
Since \(x^{2} e^{-x}\) approaches zero, it grows slower than \(x^{2}\) as \(x \to \infty\).
Conclude that \(f(x) = x^{2} e^{-x}\) grows slower than \(x^{2}\) when \(x\) becomes very large.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Growth Rates of Functions
Growth rates describe how functions behave as the input becomes very large. Comparing growth rates helps determine which functions increase faster, slower, or at the same pace as a reference function, such as x², by analyzing their dominant terms as x approaches infinity.
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04:16
Intro To Related Rates
Exponential Decay vs Polynomial Growth
Exponential decay functions like e^(-x) decrease rapidly to zero as x increases, often overpowering polynomial growth terms. When combined, such as in x²e^(-x), the exponential decay dominates, causing the overall function to approach zero faster than any polynomial grows.
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09:29Exponential Growth & Decay
Limits and Asymptotic Behavior
Evaluating limits as x approaches infinity reveals the long-term behavior of functions. By calculating the limit of the ratio of two functions, we can determine if one grows faster, slower, or at the same rate as the other, which is essential for classifying growth rates.
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5:50Asymptotes of Hyperbolas
Related Practice
Textbook Question
Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.
7. c. arcsec(-2)
Textbook Question
82. Use the definitions of the hyperbolic functions to find each of the following limits.
c. lim(x→∞) sinh x
1
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Textbook Question
Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.
5. c. arccos(√3/2)
Textbook Question
6. Which of the following functions grow faster than ln(x) as x→∞? Which grow at the same rate as ln(x)? Which grow slower?
c. 1/√x
Textbook Question
80. Find all values of c that satisfy the conclusion of Cauchy's Mean Value Theorem for the given functions and interval.
c. f(x) = x³/ (3 - 4x), g(x) = x², (a, b) = (0, 3)