Textbook Question
c. Find the slopes of the tangent lines to the graphs of f and g at (1, 1) and (−1, −1) (four tangent lines in all).
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 7.8.1.c
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?
c. √x
Verified step by step guidance1
Recall the growth rates of common functions as \(x \to \infty\). The exponential function \(e^x\) grows faster than any polynomial or root function.
Identify the function given: \(\sqrt{x}\), which is equivalent to \(x^{1/2}\), a root function and a type of power function.
Compare the growth of \(\sqrt{x}\) to \(e^x\). Since \(e^x\) grows exponentially and \(\sqrt{x}\) grows polynomially, \(e^x\) grows faster than \(\sqrt{x}\) as \(x \to \infty\).
Conclude that \(\sqrt{x}\) grows slower than \(e^x\) as \(x \to \infty\).
Summarize: \(\sqrt{x}\) does not grow faster than or at the same rate as \(e^x\); it grows slower.
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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 approaches infinity. Comparing growth rates helps determine which functions increase faster, slower, or at the same pace. For example, exponential functions like e^x grow faster than polynomial or root functions as x→∞.
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04:16
Intro To Related Rates
Exponential Functions
Exponential functions have the form a^x, where the variable is in the exponent. The function e^x is a fundamental exponential function that grows rapidly as x increases. Its growth rate surpasses that of any polynomial or root function for large x.
Recommended video:
6:13Exponential Functions
Polynomial and Root Functions
Polynomial functions involve variables raised to constant powers, while root functions are fractional powers (e.g., √x = x^(1/2)). These functions grow slower than exponential functions like e^x as x→∞, meaning their values increase but at a much slower rate.
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07:00Taylor Polynomials
Related Practice
Textbook Question
10. True, or false? As x→∞,
c. 1/x - 1/x² = o(1/x)
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:
c. Find the equation for the tangent line to f at the specified point (x_0, f(x_0)).
67. y= √(3x-2), 2/3 ≤ x ≤ 4, x_0=3
Textbook Question
In Exercises 1–4, show that each function y=f(x) is a solution of the accompanying differential equation.
1. 2y' + 3y = e^(-x)
c. y = e^(-x) + Ce^(-(3/2)x)
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?
c. √(1+x^4)
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Textbook Question
88. Given that x>0, find the maximum value, if any, of
c. x^(1/x^n) (n a positive integer)