Problem 3 - 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? $$ \begin{array}{ll}{\text { a. } x^{2}+4 x} & {\text { b. } x^{5}-x^{2}} \\\ {\text { c. } \sqrt{x^{4}+x^{3}}} & {\text { d. }(x+3)^{2}} \\ {\text { e. } x \ln x} & {\text { f. } 2^{x}} \\ {\text { g. } x^{3} e^{-x}} & {\text { h. } 8 x^{2}}\end{array} $$

Step-by-Step Solution

Answer

Faster: b, f. Same rate: a, c, d, h. Slower: e, g.

Step 1: Identify the Orders of Growth

The function orders of growth can be compared using limits. We will look at the highest power of each function's dominant term, as that determines growth as \(x \rightarrow \infty\).

Step 2: Analyze Function (a)

The function is \(x^2 + 4x\). The dominant term is \(x^2\), hence it grows at the same rate as \(x^2\).

Step 3: Analyze Function (b)

The function is \(x^5 - x^2\). The dominant term is \(x^5\), so it grows faster than \(x^2\).

Step 4: Analyze Function (c)

The function is \(\sqrt{x^4 + x^3}\). Simplifying, it becomes \(x^2 \sqrt{1 + x^{-1}} \approx x^2\) as \(x \to \infty\), meaning it grows at the same rate as \(x^2\).

Step 5: Analyze Function (d)

The function is \((x + 3)^2\). Expanding gives \(x^2 + 6x + 9\), with \(x^2\) as the dominant term, thus it grows at the same rate as \(x^2\).

Step 6: Analyze Function (e)

The function is \(x \ln x\). Although \(\ln x\) grows slowly, as \(x\to\infty\), the product with \(x\) grows slower than \(x^2\) since \(\ln x\) increases slower than any polynomial term.

Step 7: Analyze Function (f)

The function is \(2^x\). Exponential functions grow faster than any polynomial as \(x \to \infty\), so it grows faster than \(x^2\).

Step 8: Analyze Function (g)

The function is \(x^3 e^{-x}\). The exponential decay \(e^{-x}\) overtakes the polynomial growth \(x^3\), causing this function to grow slower than \(x^2\).

Step 9: Analyze Function (h)

The function is \(8 x^2\). The dominant term is \(x^2\), meaning it grows at the same rate as \(x^2\).

Key Concepts

Dominant Term

When comparing the rates at which functions grow, focusing on the dominant term is crucial. This concept helps us understand how a function behaves as the input grows larger. The dominant term is the term with the largest exponent or highest order when the function is expressed in its standard form.

• In the function, the dominant term determines the order of growth.
• It simplifies the process of comparing functions since only the highest power matters.

For example, in the polynomial function \(x^2 + 4x\), the dominant term is \(x^2\). Even though \(4x\) is added, its influence diminishes as \(x\) becomes very large. Hence, the order of growth of the function is dictated by \(x^2\). Similarly, for \(x^5 - x^2\), the dominant term is clearly \(x^5\), leading to a faster growth rate than \(x^2\). Always seek the dominant term to simplify growth analysis.

Polynomial Functions

Polynomial functions are mathematical expressions involving a sum of powers of one or more variables multiplied by coefficients. These coefficients tell us how quickly the function will grow.

• They are usually expressed as \(a_n x^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\).
• The highest power of \(x\) in a polynomial indicates the dominant term and defines its highest order of growth.

Considering the polynomial \(x^5 - x^2\), the highest power of \(x\) is 5, indicating that this polynomial grows faster than \(x^2\). For \(\sqrt{x^4 + x^3}\), we simplify it as \(x^2\), showing it grows at a similar rate as \(x^2\). Expansion of \((x+3)^2\) reveals it as \(x^2 + 6x + 9\), where \(x^2\) is dominant for large \(x\). Consequently, this function also grows at the same rate as \(x^2\) due to its highest power.

Exponential Functions

Exponential functions are characterized by having a variable in the exponent, such as \(2^x\). They show a dramatic growth pattern, often vastly outpacing polynomial growth as the input increases.

• Exponential functions grow radically faster than polynomial functions.
• They often model scenarios involving rapid increase or decrease, such as population growth or radioactive decay.

In the function \(2^x\), as \(x\) approaches infinity, the function's growth rate far exceeds any polynomial function, including \(x^2\). Another example, \(x^3 e^{-x}\), despite having a polynomial term, is overtaken by the exponential decay term \(e^{-x}\), resulting in an overall slower growth. Exponential functions differ significantly from each other based on their bases, but typically, any positive base \(b\) results in highly accelerated growth compared to polynomial terms.