Problem 50 - Precalculus : Building Concepts and Connections

Question

Find the average rate of change of each ficnetion on the given interval. $$f(x)=-x^{4}+6 x^{2}-1 ; \text { interval: }[1,2]$$

Step-by-Step Solution

Answer

The average rate of change of the function \( f(x) = -x^{4}+6x^{2}-1 \) on the interval [1,2] is -7.

Step 1: Identify the Function and Interval

The function \( f(x) = -x^{4}+6x^{2}-1 \) and the interval is [1,2]. That is, \(a = 1\) and \(b = 2\).

Step 2: Evaluate the Function at the Endpoints of the Interval

Calculate \(f(a)\) and \(f(b)\) by substituting \(a = 1\) and \(b = 2\) into the function. This gives \(f(a) = -1^{4}+6*1^{2}-1 = 4 \) and \(f(b) = -2^{4}+6*2^{2}-1 = -3\).

Step 3: Apply the Average Rate of Change Formula

Use the formula for average rate of change: \( \frac{f(b) - f(a)}{b - a} \). Substitute the calculated \(f(a)\) and \(f(b)\) values, and the known \(a\) and \(b\) values into the formula, yielding \( \frac{-3 - 4}{2 - 1} = -7 \).

Key Concepts

Understanding Precalculus

Precalculus is a course that prepares students for calculus, the mathematical study of change. It encompasses a variety of topics that form the groundwork for understanding advanced concepts in calculus. Subjects covered in precalculus include algebraic expressions, functions, trigonometry, and the concept of limits, among others.

In the context of function analysis, precalculus involves the study of polynomial functions and their properties. Recognizing how these functions behave, and how rates of change can be calculated, is essential for students looking to advance their mathematical skills. By strengthening their foundation in precalculus, students can better understand and tackle the complexities of calculus.

Demystifying Polynomial Functions

Polynomial functions are algebraic expressions that consist of variables raised to whole number exponents and their coefficients. In precalculus, much emphasis is placed on understanding these functions because they appear in various forms and complexities. For example, a quadratic function is a second-degree polynomial, and a function of the form \( f(x) = -x^4 + 6x^2 - 1 \) is a fourth-degree polynomial.

Key features of polynomial functions that are often studied include their:

• End behavior, determined by the highest power term
• Zeroes or roots, where the function crosses the x-axis
• Turning points, which are local maxima and minima
• Intervals of increase and decrease

Understanding the characteristics of polynomial functions enables students to predict their graphical behavior and solve complex problems involving rates of change.

Rate of Change Calculation

The rate of change is fundamental in mathematics as it describes how a quantity changes over time or across different values. In terms of functions, the average rate of change between two points provides insight into the function's behavior over that interval. It is calculated using the formula: \( \frac{f(b) - f(a)}{b - a} \), where \(a\) and \(b\) are points on the domain and \(f(a)\) and \(f(b)\) are the corresponding points on the range.

This calculation gives rise to the concept of slope in linear functions, but is also equally vital in understanding the behavior of polynomial functions. For instance, finding the average rate of change for \( f(x) = -x^4 + 6x^2 - 1 \) from \(x=1\) to \(x=2\) will indicate how fast the function's output is changing over that interval. A negative average rate of change, as seen in the exercise, suggests that the function decreases as \(x\) moves from 1 to 2. Understanding this concept helps students to discern function behavior efficiently without having to graph it.