Problem 6 - Calculus

Question

In Exercises \(3-10\), determine the end behavior of each function as \(x \rightarrow+\infty\) and as \(x \rightarrow-\infty\). $$f(x)=1000-38 x+50 x^{2}-5 x^{3}$$

Step-by-Step Solution

Answer

As \(x \rightarrow +\infty\), \(f(x) \rightarrow -\infty\); as \(x \rightarrow -\infty\), \(f(x) \rightarrow +\infty\).

Step 1: Identify the Leading Term

The function given is a polynomial: \(f(x) = 1000 - 38x + 50x^2 - 5x^3\). To determine the end behavior, we focus on the leading term, which is the term with the highest power of \(x\). In this polynomial, the leading term is \(-5x^3\), as it has the highest degree (3).

Step 2: Analyze End Behavior as \(x \rightarrow +\infty\)

For polynomials, the leading term dictates the end behavior. Since the leading coefficient is \(-5\) (a negative number) and the exponent is 3 (an odd number), as \(x \rightarrow +\infty\), \(-5x^3\rightarrow -\infty\). Thus, \(f(x) \rightarrow -\infty\).

Step 3: Analyze End Behavior as \(x \rightarrow -\infty\)

Considering the same leading term \(-5x^3\), when \(x \rightarrow -\infty\), \((-5)(-x)^3 \rightarrow +\infty\) due to the odd exponent reversing the sign again. Therefore, \(f(x) \rightarrow +\infty\) as \(x \rightarrow -\infty\).

Key Concepts

Leading Term

In a polynomial, identifying the leading term is crucial for understanding the behavior of the function as the input approaches very large (positive or negative) values. The leading term of a polynomial is simply the term with the highest exponent of the variable. When we analyze functions mathematically, this term becomes the most significant in determining the long-range behavior of the function.
For example, in the polynomial \(f(x) = 1000 - 38x + 50x^2 - 5x^3\), the leading term is found by locating the term with the highest power, which is \(-5x^3\). This term dominates the function's behavior as \(x\) moves toward infinity, either positive or negative.

• The leading coefficient here is \(-5\).
• The leading exponent is \(3\), which is odd.

Knowing these two characteristics allows us to predict how the function behaves at the extremes.

Polynomials

Polynomials are algebraic expressions that consist of terms in the form \(ax^n\), where \(a\) is the coefficient, \(x\) is the variable, and \(n\) is a non-negative integer exponent. These expressions can have a single term or several, and they are often represented in a descending order of exponents.
The general form of a polynomial is given by:

• \(P(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\)
• Where \(a_n eq 0\), making \(a_nx^n\) the leading term.

Polynomials are versatile and appear in various areas of mathematics due to their predictable behavior and gentle slopes. When analyzing polynomials, particularly for determining end behavior, we examine how the function values change as \(x\) approaches either positive or negative infinity. Understanding polynomials allows us to comprehend these transitions smoothly.

Infinite Limits

Infinite limits describe the behavior of functions as the inputs approach either positive or negative infinity. This behavior is especially important in understanding the end behavior of polynomial functions, where we check the influence of the leading term as \(x\) moves toward infinity.
For a polynomial like \(-5x^3\), the negative coefficient and odd exponent tells us:

• As \(x \rightarrow +\infty\), \(-5x^3 \rightarrow -\infty\)
• As \(x \rightarrow -\infty\), \(-5x^3 \rightarrow +\infty\)

These directions of growth, or divergence, arise because with an odd power, the sign of \(x\) is preserved if \(x\) is negative, but reverses the sign when multiplied by a negative leading coefficient when \(x\) is positive. Infinite limits help in predicting polynomial's behavior far from the easily calculable values, making them valuable in planning and predictions in fields like physics, engineering, and economics.