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 7.8.10.a
10. True, or false? As x→∞,
a. 1/(x+3) = O(1/x)
Verified step by step guidance1
Recall the definition of Big-O notation: A function \(f(x)\) is \(O(g(x))\) as \(x \to \infty\) if there exist positive constants \(C\) and \(x_0\) such that for all \(x > x_0\), \(|f(x)| \leq C |g(x)|\).
Identify the functions in the problem: \(f(x) = \frac{1}{x+3}\) and \(g(x) = \frac{1}{x}\).
Compare the behavior of \(f(x)\) and \(g(x)\) as \(x \to \infty\). Notice that \(\frac{1}{x+3}\) behaves very similarly to \(\frac{1}{x}\) because adding 3 becomes insignificant for very large \(x\).
To verify the Big-O relationship, consider the ratio \(\frac{|f(x)|}{|g(x)|} = \frac{\frac{1}{x+3}}{\frac{1}{x}} = \frac{x}{x+3}\). As \(x \to \infty\), this ratio approaches 1, which is a finite constant.
Since the ratio approaches a finite constant, there exist constants \(C\) and \(x_0\) such that \(|f(x)| \leq C |g(x)|\) for all \(x > x_0\), confirming that \(\frac{1}{x+3} = O\left(\frac{1}{x}\right)\) is true.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Big O Notation
Big O notation describes the upper bound of a function's growth rate, focusing on its behavior as the input approaches infinity. It provides a way to classify functions according to their dominant terms, ignoring constant factors and lower-order terms.
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Sigma Notation
Asymptotic Behavior of Functions
Asymptotic behavior studies how functions behave as the input variable grows very large. Understanding limits and dominant terms helps compare functions like 1/(x+3) and 1/x as x approaches infinity.
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Equivalence of Functions in Big O Terms
Two functions f(x) and g(x) are related by f(x) = O(g(x)) if there exist constants C and x0 such that |f(x)| ≤ C|g(x)| for all x > x0. This means f(x) grows no faster than a constant multiple of g(x) for large x.
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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. 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
143.
a. Show that ∫ ln(x) dx = x ln(x) − x + C.
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?
a. x-3
1
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Textbook Question
154. The linearization of log₃x
a. Find the linearization of
f(x) = log₃xatx = 3.
Then round its coefficients to two decimal places.