Problem 5 - Calculus

Question

In Exercises \(3-10\), determine the end behavior of each function as \(x \rightarrow+\infty\) and as \(x \rightarrow-\infty\). $$f(x)=5 x^{4}-25 x^{3}-62 x^{2}+5 x+300$$

Step-by-Step Solution

Answer

As \(x \rightarrow \pm \infty\), \(f(x) \rightarrow +\infty\).

Step 1: Identify Leading Term

The given polynomial function is \(f(x) = 5x^4 - 25x^3 - 62x^2 + 5x + 300\). To determine the end behavior, we focus on the leading term of the polynomial, which is the term with the highest power of \(x\). In this case, the leading term is \(5x^4\).

Step 2: Analyze Leading Term as \(x \rightarrow +\infty\)

The leading term \(5x^4\) dominates the behavior of \(f(x)\) as \(x\) becomes very large positively. Since \(5x^4\) is a positive term and \(x^4\) grows without bounds, \(f(x)\) will also grow without bounds. Thus, as \(x \rightarrow +\infty\), \(f(x) \rightarrow +\infty\).

Step 3: Analyze Leading Term as \(x \rightarrow -\infty\)

As \(x\) becomes very large negatively, the leading term \(5x^4\) is still the main factor in determining the behavior of \(f(x)\). Since \((x^4)\) is positive for both positive and negative \(x\), and \(5\) is positive, \(f(x)\) will grow positively. Thus, as \(x \rightarrow -\infty\), \(f(x) \rightarrow +\infty\).

Key Concepts

Leading Term

In a polynomial function, the leading term is the term with the highest power of the variable, which is typically represented as \(x\). This term is crucial in determining the end behavior of the polynomial function because it has the greatest influence on the function's value as \(x\) becomes extremely large or extremely small.
For instance, in the polynomial \(f(x) = 5x^4 - 25x^3 - 62x^2 + 5x + 300\), the leading term is \(5x^4\).

• The exponent of 4 in \(5x^4\) tells us that it is a fourth-degree polynomial.
• The positive coefficient, 5, indicates that as \(x\) tends towards both positive and negative infinity, \(5x^4\) will influence \(f(x)\) by growing incrementally positive.

Understanding which term is the leading term helps simplify the analysis of a polynomial function's behavior across its domain.

Polynomial Function

A polynomial function is a mathematical expression involving a sum of powers of a variable, each multiplied by coefficients. These functions are typically expressed in the form \(a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0\), where \(a_n, a_{n-1}, ..., a_0\) are constants and \(n\) is a non-negative integer.

Here are some key features of polynomial functions:

• The degree of the polynomial is the highest power of the variable in the expression, which determines the number of roots and the potential number of turns in its graph.
• Polynomial functions can have a wide variety of behaviors and shapes, but they are always smooth and continuous.
• With integer exponents, they never have sharp corners or breaks.

Understanding these features can help in predicting and analyzing how a polynomial function behaves over different intervals.

End Behavior Analysis

End behavior analysis involves predicting what happens to the value of a function as the input \(x\) becomes very large or very small. Essentially, it's about understanding how the function "behaves" at its extremes.

To analyze end behavior of a polynomial function:

• Identify the leading term, as it will dominate the behavior of the function for very large or very small values of \(x\).
• Consider the coefficient and the power of \(x\) in the leading term, because these dictate how the function rises or falls.
• For even powers like \(x^4\) in our example, the sign of the coefficient determines whether both ends of the graph point up (positive) or down (negative).
• In our example of \(f(x) = 5x^4 - 25x^3 - 62x^2 + 5x + 300\), the positive leading term \(5x^4\) means \(f(x)\) trends towards \(+\infty\) as \(x\to +\infty\) and \(x\to -\infty\).

By focusing on these points, students can more easily predict the end behavior of a polynomial function.