Question

Find the average rate of change \(f(x)=x^{3}\) from -1 to 3 .

Short answer

The average rate of change is 7.

Step-by-step solution

- Understand the formula for Average Rate of Change

The average rate of change of a function over an interval \([a, b]\) is given by the formula: \[\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}\] For the function \(f(x) = x^3\) and the interval \([-1, 3]\), identify \a = -1\ and \b = 3\.

- Evaluate the function at the endpoints

Substitute \a = -1\ and \b = 3\ into \(f(x) = x^3\) to find \(f(a)\) and \(f(b)\).\[f(-1) = (-1)^3 = -1\]\[f(3) = 3^3 = 27\]

- Calculate the Average Rate of Change

Now apply the values obtained into the average rate of change formula: \[\text{Average Rate of Change} = \frac{f(3) - f(-1)}{3 - (-1)} = \frac{27 - (-1)}{3 + 1} = \frac{27 + 1}{4} = \frac{28}{4} = 7\]

Key concepts

Interval Notation

Interval notation is a way of representing a set of numbers between two endpoints. These sets are crucial in calculus and algebra for defining the domain of functions, integrals, and more.
To denote an interval:
* \([a, b]\) indicates all numbers between and including \(a\) and \(b\).
* For example, interval \([-1, 3]\) includes −1, 3, and every value in between. Such ranges make understanding where a function is evaluated easier.

Cubic Functions

A cubic function is a polynomial of degree three, typically written in the form \(f(x) = ax^3 + bx^2 + cx + d\). These functions can have up to three real roots and can change direction twice.
The general shape of a cubic function can have a point of inflection, where the concavity changes, making these curves more interesting than linear or quadratic functions.
For example, \(f(x) = x^3\) rises steeply and continuously for both positive and negative values of \(x\). This behavior makes it simpler when performing evaluations at various points.

Rate of Change Formula

The rate of change formula quantifies how one quantity changes, on average, relative to another. For a function \(f(x)\), its average rate of change over an interval \([a, b]\) can be found by:
\[ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} \]
This formula essentially calculates the slope of the line connecting two points \( (a, f(a)) \) and \( (b, f(b)) \) on the function’s graph.
Using the earlier example, for \(f(x) = x^3\) from −1 to 3, the average rate of change was found by substituting the function values into this formula.

Function Evaluation

Function evaluation involves finding the output of a function for specific input values. To evaluate a function \(f(x) \), you substitute the input \(x\) value into the function.
For instance, with \(f(x) = x^3\):
* Evaluating at \( x = -1 \) yields \( f(-1) = (-1)^3 = -1 \).
* Evaluating at \( x = 3 \) yields \( f(3) = 3^3 = 27 \).
These evaluations are essential for determining the behavior of the function over an interval and for calculating quantities like the average rate of change.