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Ch. 7 - Transcendental Functions
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 7 - Transcendental FunctionsProblem 7.8.9.c
Chapter 7, Problem 7.8.9.c

9. True, or false? As x→∞,
c. x = O(x+5)

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1
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: Here, \(f(x) = x\) and \(g(x) = x + 5\).
Analyze the behavior of \(f(x)\) and \(g(x)\) as \(x \to \infty\): Since \(x + 5\) behaves like \(x\) for large \(x\), the two functions grow at the same rate.
Set up the inequality to check if \(x = O(x + 5)\): We want to find constants \(C\) and \(x_0\) such that \(x \leq C (x + 5)\) for all \(x > x_0\).
Consider dividing both sides by \(x + 5\) (which is positive for large \(x\)) to get \(\frac{x}{x + 5} \leq C\). Since \(\lim_{x \to \infty} \frac{x}{x + 5} = 1\), we can choose \(C\) slightly larger than 1 and find an \(x_0\) to satisfy the inequality.

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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 an upper bound on the growth rate of a function as its input approaches infinity. It is used to compare the asymptotic behavior of functions, indicating that one function grows no faster than another up to a constant multiple.
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Sigma Notation

Asymptotic Behavior of Functions

Asymptotic behavior studies how functions behave as the input becomes very large. Understanding this helps determine if one function can be bounded by another, which is essential for applying Big O notation correctly.
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Asymptotes of Hyperbolas

Comparison of Linear Functions

When comparing linear functions like x and x+5, the constant term becomes insignificant as x approaches infinity. This means x and x+5 grow at the same rate asymptotically, which is key to evaluating statements involving Big O notation.
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Related Practice
Textbook Question

Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.

1. c. tan^(-1)(1/√3)

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)).


70. y= x³/(x²+1), -1 ≤ x ≤ 1, x_0=1/2

Textbook Question

Suppose that the function f and its derivative with respect to x have the following values at x=0, 1, 2, 3, and 4.

Assuming the inverse function f^(-1) is differentiable, find the slope of f^(-1)(x) at

c. x=3

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).

Textbook Question

In Exercises 41–44:


c. Evaluate df/dx at x = a and df⁻¹/dx at x = f(a) to show that

(df⁻¹/dx)|ₓ₌f(a) = 1 / (df/dx)|ₓ₌a


44. f(x) = 2x², x ≥ 0, a = 5

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)).

68. y= (3x+2)/(2x-11), -2 ≤ x ≤ 2, x_0=1/2