Problem 4 - Thomas Calculus

Question

Which of the following functions grow faster than \(x^{2}\) as \(x \rightarrow \infty ?\) Which grow at the same rate as \(x^{2} ?\) Which grow slower? $$ \begin{array}{ll}{\text { a. } x^{2}+\sqrt{x}} & {\text { b. } 10 x^{2}} \\\ {\text { c. } x^{2} e^{-x}} & {\text { d. } \log _{10}\left(x^{2}\right)} \\\ {\text { e. } x^{3}-x^{2}} & {\text { f. }(1 / 10)^{x}} \\ {\text { g. }(1.1)^{x}} & {\text { h. } x^{2}+100 x}\end{array} $$

Step-by-Step Solution

Answer

a, b, h grow at the same rate; e, g grow faster; c, d, f grow slower.

Step 1: Analyze Function a.

Consider the function \( f_a(x) = x^2 + \sqrt{x} \). The term \( x^2 \) is the dominant term because \( \sqrt{x} \) grows slower than \( x^2 \). As \( x \to \infty \), the \( x^2 \) term dominates, so \( f_a(x) \) grows at the same rate as \( x^2 \).

Step 2: Analyze Function b.

Consider the function \( f_b(x) = 10x^2 \). The function is a constant multiple of \( x^2 \), so it grows at the same rate as \( x^2 \).

Step 3: Analyze Function c.

Consider the function \( f_c(x) = x^2 e^{-x} \). The term \( e^{-x} \) decreases exponentially as \( x \to \infty \), causing the whole function to approach zero. Therefore, \( f_c(x) \) grows slower than \( x^2 \).

Step 4: Analyze Function d.

Consider the function \( f_d(x) = \log_{10}(x^2) \). This can be rewritten as \( 2\log_{10}(x) \), which grows much slower than \( x^2 \) since logarithmic growth is slower than polynomial growth.

Step 5: Analyze Function e.

Consider the function \( f_e(x) = x^3 - x^2 \). The \( x^3 \) term dominates as \( x \to \infty \), so \( f_e(x) \) grows faster than \( x^2 \) because cubic growth outpaces quadratic growth.

Step 6: Analyze Function f.

Consider the function \( f_f(x) = (1/10)^x \). This function is an exponential decay, so it approaches zero as \( x \to \infty \), growing slower than \( x^2 \).

Step 7: Analyze Function g.

Consider the function \( f_g(x) = (1.1)^x \). This is an exponential growth function, which grows faster than any polynomial, including \( x^2 \).

Step 8: Analyze Function h.

Consider the function \( f_h(x) = x^2 + 100x \). The \( x^2 \) term is the dominant term, so \( f_h(x) \) grows at the same rate as \( x^2 \).

Key Concepts

Function Growth Rates

When we talk about function growth rates, we're discussing how fast a function increases as its input becomes large. The idea is to compare how different expressions behave as their variable approaches infinity. In this exercise, we explore which functions grow faster, slower, or at the same rate compared to a quadratic function, specifically, \( x^2 \).

To analyze growth rates, we focus on the term within the function that increases most rapidly as \( x \) grows larger. This leading term can determine the overall behavior of the function's growth. For example:

• Cubic functions (like \( x^3 \)) grow faster than quadratic functions because they have an extra factor of \( x \).
• Linear functions (like \( x \)) grow slower than quadratics since \( x^1 \) can't keep up with \( x^2 \).
• Constant multiples (like \( 10x^2 \)) grow at the same rate as quadratics, as the multiplication by a constant does not affect the growth behavior.

Understanding function growth rates is crucial when ranking algorithms or determining efficiency in computational problems.

Exponential Growth and Decay

Exponential functions have growth or decay patterns that significantly differ from polynomial ones. They are expressed in the form \( a^x \), where \( a > 1 \) for growth, and \( 0 < a < 1 \) for decay.

Exponential growth is incredibly rapid because as \( x \) increases, the function values increase at a rate proportional to their current size. An example is \( (1.1)^x \). As \( x \) grows, \( (1.1)^x \) eventually surpasses polynomial growth like \( x^2 \).

Conversely, exponential decay, such as \( (1/10)^x \), shrinks very quickly and approaches zero as \( x \) increases. This means it decreases much faster than polynomial terms could increase.

When comparing functions, it's essential to consider these exponential patterns, as they either skyrocket to infinity rapidly or diminish to nothing, showcasing different efficiency and behavior in practical applications.

Logarithmic Functions

Logarithmic functions, indicated as \( \log(x) \), grow much slower than any polynomial function. They scale as the inverse of exponential functions and are crucial in scenarios where very slow growth is desired or significant.

In our exercise, we consider \( \log_{10}(x^2) \), which can be simplified to \( 2 \log_{10}(x) \). Even though the square inputs seem substantial, logarithmic growth means the output increases at a decreasing rate. This makes them slower than quadratic growth.

Logarithmic growth often appears in algorithms, particularly in data structures and processes using divide and conquer techniques where reductions by half are common. Understanding these functions are critical in computing fields, as they often point to more efficient solutions or minimal resource usage over substantial scales.

By comprehending the nature of logarithmic functions, you can better appreciate their unique slow growth and practical application in problem-solving.