Problem 41 Rank the functions in order of h... [FREE SOLUTION]

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Biocalculus Calculus for the Life Sciences

James Stewart

$Math Studyset 91影视 Explanations$ Math

1 Edition

Chapter 4: Problem 41

Rank the functions in order of how quickly they grow as $x \rightarrow \infty$. $y=(\ln x)^{2}, \quad y=(\ln x)^{3}, \quad y=\sqrt{x}, \quad y=\sqrt[3]{x}$

Short Answer

Expert verified

$ y = \\sqrt{x} > y = \\sqrt[3]{x} > y = (\ln x)^3 > y = (\ln x)^2 $.

Step by step solution

Analyze Polynomial Growth

Consider the function $ y = x^n $. If a function is a polynomial like $ y = \sqrt{x} = x^{1/2} $ or $ y = \sqrt[3]{x} = x^{1/3} $, we compare their powers. Here, $ y = x^{1/2} $ grows faster than $ y = x^{1/3} $ because $ 1/2 > 1/3 $.

Analyze Logarithmic Growth

Consider logarithmic functions such as $ y = (\ln x)^2 $ and $ y = (\ln x)^3 $. Since $ (\ln x)^3 $ contains a higher power of the logarithmic term than $ (\ln x)^2 $, $ y = (\ln x)^3 $ grows faster than $ y = (\ln x)^2 $ as $ x \rightarrow \infty $.

Compare Polynomial and Logarithmic Growth

Polynomial functions like powers of $ x $ (e.g., $ y = \sqrt{x} $) generally grow faster than logarithmic functions raised to a power (e.g., $ y = (\ln x)^2 $ or $ y = (\ln x)^3 $) as $ x \rightarrow \infty $. Hence, $ y = \sqrt{x} $ and $ y = \sqrt[3]{x} $ both grow faster than $ y = (\ln x)^2 $ and $ y = (\ln x)^3 $.

Rank the Functions

Combine the insights: the fastest-growing function is $ y = \sqrt{x} $, followed by $ y = \sqrt[3]{x} $, then $ y = (\ln x)^3 $, and the slowest-growing is $ y = (\ln x)^2 $. The rank order is: $ y = \sqrt{x} > y = \sqrt[3]{x} > y = (\ln x)^3 > y = (\ln x)^2 $.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polynomial Functions

Polynomial functions are essential building blocks in mathematics. They are expressions involving a variable raised to a power, such as $ y = x^n $. The term "power" here refers to the exponent $ n $, which can be any real number. Polynomials can have various characteristics influencing their growth:

The higher the exponent, the faster the polynomial grows as $ x $ approaches infinity. For instance, comparing $ y = x^{1/2} $ and $ y = x^{1/3} $, the former grows faster because 1/2 is greater than 1/3.
Polynomial functions have a straightforward asymptotic behavior, as they steadily increase without bound when the exponent is positive.

Polynomial growth is generally more rapid than logarithmic growth, which we'll explore next. Understanding polynomial functions is crucial as they form the basis for calculating rates of growth, analyzing trends, and solving complex mathematical problems.

Logarithmic Functions

Logarithmic functions involve exponents and growth differently than polynomials. A basic example is the natural logarithm, denoted as $ \ln x $. When raised to a power, as in $ y = (\ln x)^n $, they reflect slow growth patterns:

Higher powers of logarithmic functions, such as $ (\ln x)^3 $ versus $ (\ln x)^2 $, still grow more slowly than comparable polynomial functions with fractional powers.
Logarithms grow by increasing small incremental values. For example, transitioning from $ \ln 10 $ to $ \ln 100 $ represents a small growth step in a logarithm's entire range.

Understanding logarithmic growth helps in fields like algorithm analysis, where such functions frequently describe time complexity, providing insights into computational efficiency. Despite their slower growth compared to polynomials, they are essential in describing naturally occurring phenomena and processes.

Asymptotic Behavior

Asymptotic behavior provides insight into how functions grow relative to each other as $ x $ approaches infinity. It helps identify which functions grow faster as they extend towards larger values of $ x $. Here are some key points:

Polynomials typically display straightforward asymptotic behavior due to their constant faster growth, especially compared to logarithmic functions.
Logarithmic functions grow even more slowly, showing only slight increases even as $ x $ becomes very large.
In asymptotic terms, the ranking of the growth rates for our original set of functions was derived: $ y = \sqrt{x} $ grows fastest, followed by $ y = \sqrt[3]{x} $, then $ y = (\ln x)^3 $, and lastly, $ y = (\ln x)^2 $.

Understanding asymptotic behavior is pivotal in mathematics because it helps in approximating and simplifying functions without detailed calculations. It's especially useful in fields like computer science, where performance predictions often rely on determining which algorithm will execute more efficiently at large inputs.

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91影视

Rank the functions in order of how quickly they grow as \(x \rightarrow \infty\). \(y=(\ln x)^{2}, \quad y=(\ln x)^{3}, \quad y=\sqrt{x}, \quad y=\sqrt[3]{x}\)

Short Answer

Step by step solution

Analyze Polynomial Growth

Analyze Logarithmic Growth

Compare Polynomial and Logarithmic Growth

Rank the Functions

Key Concepts

Polynomial Functions

Logarithmic Functions

Asymptotic Behavior

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