Problem 46 - Precalculus : Building Concepts and Connections

Question

Find the average rate of change of each ficnetion on the given interval. $$f(x)=x^{3}+1 ; \text { interval: }[0,2]$$

Step-by-Step Solution

Answer

The average rate of change of the function \(f(x) = x^3 + 1\) over the interval [0,2] is 4

Step 1: Evaluate the function at the endpoints

Evaluate the function \(f(x) = x^3 + 1\) at the endpoints of the interval [0,2]. We obtain \(f(0) = 0^3+1 = 1\) and \(f(2) = 2^3+1 = 9\).

Step 2: Calculate change

Calculate the change in the output (ΔY) and input (ΔX). ΔY is the change in function from \(f(2)\) to \(f(0)\), which is \(9-1 = 8\). ΔX is the change of x-values (2-0) is 2.

Step 3: Find Average Rate

The average rate of change of the function over the interval is the change in output divided by the change in input. It is therefore, \(\frac{ΔY}{ΔX} = \frac{8}{2} = 4\).

Key Concepts

Calculus

Calculus is a branch of mathematics that studies how things change. It is a fundamental part of modern mathematical education, focusing on concepts like derivatives and integrals. These concepts help us understand how rates of change operate in different contexts. For example, when studying the average rate of change in a function, calculus allows us to analyze how a small change in one variable influences another. This is crucial for science and engineering, where we often need to predict future events based on past trends.

Functions

A function describes how one quantity changes with respect to another. In simple terms, it's a relationship between two variables, usually represented as \( f(x) \) in mathematical language. This allows us to denote the output of a function, \( y \), for any given input \( x \). Functions can be linear, quadratic, cubic, and more complex, describing a wide array of natural and human-made phenomena.

• Linear functions have constant rates of change.
• Cubic functions like \( x^3 + 1 \) can vary significantly as \( x \) changes.

Understanding the type of function used helps predict how the quantity of interest will behave as changes are made to the variables.

Interval Evaluation

Interval evaluation is the process of assessing a function over a specific range of values. This is crucial for understanding how a function behaves between two points. In our exercise, we look at the function \( f(x) = x^3 + 1 \) over the interval \([0, 2]\). By evaluating the function at these endpoints, we can observe how the function's output changes over this specific interval.

• The output at \( x=0 \) is 1.
• The output at \( x=2 \) is 9.

These values help us calculate the average rate of change and gain insights into the overall trend between defined start and end points.

Step by Step Solutions

Using step-by-step solutions allows students to break down a problem into manageable parts, making it easier to understand complex concepts. For finding the average rate of change, we follow a systematic method:

• First, evaluate the function at the designated endpoints.
• Calculate the difference in function values, \( \Delta Y \), and the difference in \( x \)-values, \( \Delta X \).
• Your final step is to divide \( \Delta Y \) by \( \Delta X \) to find the average rate of change—essentially the 'slope' of the line connecting the two points on the function.

This organized approach not only reveals the average rate but also reinforces understanding by highlighting relationships between different aspects of a mathematical problem.