Problem 47 - Precalculus : Building Concepts and Connections

Question

Find the average rate of change of each ficnetion on the given interval. $$f(x)=-2 x^{3} ; \text { interval: }[-2,0]$$

Step-by-Step Solution

Answer

The average rate of change of the function \(f(x) = -2x^3\) on the interval \([-2,0]\) is -8.

Step 1: Substitute the Value at the Lower Endpoint

Start by substituting \(x = -2\) into \(f(x) = -2x^3\) to find the function value at the lower endpoint of the interval. The result is \(f(-2) = -2(-2)^3 = -2(-8) = 16\).

Step 2: Substitute the Value at the Upper Endpoint

Next, substitute \(x = 0\) into \(f(x) = -2x^3\) to find the function value at the upper end of the interval. The result is \(f(0) = -2(0)^3 = 0\).

Step 3: Calculate the Average Rate of Change

Substituting the function values into the formula for the average rate of change, \((f(b) - f(a))/(b - a)\), the result is \((f(0) - f(-2))/(0 - (-2)) = (0 - 16)/2 = -8\).

Key Concepts

Rate of Change

Understanding the rate of change is crucial for analyzing functions and their behaviors. In precalculus, the average rate of change is a concept that measures how much a function's output (the value of f(x)) changes between two points on its graph. This is analogous to calculating the slope of the secant line that intersects the graph at these two points.

The average rate of change can be found using the formula:
\[ \frac{f(b) - f(a)}{b - a} \]
where a and b are the x-values of the two points, and f(a) and f(b) are the function values at these points. By computing this ratio, one can gauge how fast or slow the function is changing on average over the specified interval.

Polynomial Functions

Polynomial functions are algebraic expressions that involve sums of powers of a variable. They are one of the most fundamental types of functions in mathematics and can range from simple linear functions to more complex higher-degree equations.

Formally, a polynomial function in one variable x is written as:
\[ P(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 \]
where n is a non-negative integer, and the coefficients a_n, a_{n-1}, ... , a_0 are constants. The highest power of x determines the degree of the polynomial. In the exercise, f(x) = -2x^3, this function is a third-degree polynomial, having its highest exponent as 3.

Function Intervals

When working with functions, especially when calculating rate of change, it's important to consider the interval of interest. An interval is a set of numbers between two endpoints, where we analyze the function's behavior. In the context of the exercise, the interval [-2, 0] represents all the x-values from -2 to 0, inclusive.

Intervals can be open (excluding endpoints) or closed (including endpoints), and they are crucial when discussing concepts like continuity, increasing/decreasing functions, and maximum/minimum values. Choosing the appropriate interval is intrinsic to correctly applying the average rate of change formula.