Textbook Question
80. Find all values of c that satisfy the conclusion of Cauchy's Mean Value Theorem for the given functions and interval.
a. f(x) = x, g(x) = x², (a, b) = (-2, 0)
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.a
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?
a. x-3
Verified step by step guidance1
Recall that the function \(e^x\) is an exponential function, which grows faster than any polynomial function as \(x \to \infty\).
Compare the given function \(x - 3\) to \(e^x\): since \(x - 3\) is a linear polynomial, it grows much slower than \(e^x\) as \(x\) becomes very large.
To determine growth rates, consider the limit \(\lim_{x \to \infty} \frac{f(x)}{e^x}\) for the function \(f(x) = x - 3\).
Evaluate the limit \(\lim_{x \to \infty} \frac{x - 3}{e^x}\). If this limit is 0, then \(x - 3\) grows slower than \(e^x\).
Since the limit tends to 0, conclude that \(x - 3\) grows slower than \(e^x\) as \(x \to \infty\).
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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, polynomial functions grow slower than exponential functions like e^x as x→∞.
Recommended video:
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 grows rapidly and dominates polynomial and logarithmic functions as x approaches infinity. Understanding e^x is key to comparing growth rates.
Recommended video:
6:13Exponential Functions
Limits and Asymptotic Behavior
Limits describe the behavior of functions as the input approaches a particular value, often infinity. Asymptotic behavior focuses on how functions compare in growth by examining the limit of their ratio. This helps classify functions as growing faster, slower, or at the same rate.
Recommended video:
5:50Asymptotes of Hyperbolas
Related Practice
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?
a. log_2(x²)
Textbook Question
4. Which of the following functions grow faster than x² as x→∞? Which grow at the same rate as x²? Which grow slower?
a. x² + √x
Textbook Question
4. Use the properties of logarithms to write the expressions in Exercises 3 and 4 as a single term.
a. ln secθ + ln cosθ
Textbook Question
154. The linearization of log₃x
a. Find the linearization of
f(x) = log₃xatx = 3.
Then round its coefficients to two decimal places.
Textbook Question
10. True, or false? As x→∞,
a. 1/(x+3) = O(1/x)