Problem 45 - Precalculus : Building Concepts and Connections

Question

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

Step-by-Step Solution

Answer

The average rate of change of the function \(f(x) = 3x^2 - 1\) over the interval [2, 3] is 15.

Step 1: Compute \(f(3)\) and \(f(2)\)

The first thing to do is to substitute \(x = 3\) and \(x = 2\) into the given function \(f(x) = 3x^2 - 1\). \nThis gives us \(f(3) = 3*3^2 - 1 = 26\) and \(f(2) = 3*2^2 - 1 = 11\).

Step 2: Apply the formula for the average rate of change

Then, we use the formula for the average rate of change, which is \(\frac{f(b) - f(a)}{b - a}\), substituting \(a = 2\), \(b = 3\), \(f(2) = 11\), and \(f(3) = 26\) as per our previous step. This gives us: \(\frac{f(3) - f(2)}{3 - 2} = \frac{26 - 11}{3 - 2} = 15\).

Step 3: Interpret the result

The result, 15, is the average rate of change of the function \(f(x) = 3x^2 - 1\) over the interval [2, 3]. This value describes how much our function changes, on average, for each unit increase in \(x\) over our interval.

Key Concepts

Understanding Quadratic Functions

Quadratic functions are a fundamental part of algebra and mathematics education. These functions are represented by the general form \(f(x) = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. In the exercise, the quadratic function given is \(f(x) = 3x^2 - 1\).

Quadratic functions create a parabolic curve on a graph. This means that when you plot them, they have a distinct 'U' shape, either opening upwards or downwards depending on the coefficient \(a\). If \(a\) is positive, like in our example \(3x^2 - 1\), the parabola opens upward. Quadratic functions are important because they model many real-world phenomena, from the trajectory of a thrown ball to the business world where profit maximization is considered.

• The vertex of this parabola represents the maximum or minimum point.
• The axis of symmetry is the vertical line that passes through the vertex.
• Quadratic functions are applicable in areas like physics, engineering, and economics.

Breaking Down Intervals

An interval in mathematics denotes a specific segment of the number line. In this context, the interval \([2, 3]\) means we are examining how the quadratic function changes between \(x = 2\) and \(x = 3\).

Intervals are crucial when we're trying to understand how functions behave over specific ranges of input values. By focusing on an interval, like \([2, 3]\), we can determine the rate at which a function's output changes. This is akin to checking the speed of a car over a specific distance rather than its entire journey.

• Closed intervals, denoted by brackets \([ ]\), include their endpoints.
• Open intervals, denoted by parentheses \(( )\), do not include endpoints.
• Intervals allow us to calculate average rates and other statistical measures for that segment.

Function Evaluation Made Easy

Function evaluation is a straightforward process that involves substituting specific values into a function to ascertain the output. It's a bit like testing a recipe by adding certain ingredients to see what flavor you get.

For the function \(f(x)=3x^2-1\), the task was to evaluate it at \(x = 2\) and \(x = 3\). This means replacing \(x\) with each of these values in the formula, yielding \(f(2) = 3(2)^2 - 1 = 11\) and \(f(3) = 3(3)^2 - 1 = 26\). By calculating these values, we find the outputs or the 'flavor' of the function at specific input 'ingredients'.

• Always follow order of operations: parenthesis, exponents, multiplication, division, addition, and subtraction.
• Check your work to avoid common human errors, like incorrect arithmetic.
• Function evaluation prepares the ground for more complex topics like calculus.

Simplified Mathematics Education Approach

Mathematics education aims to build a coherent understanding of fundamental concepts that help solve real-world problems. Instructors and educational content aim to break down complex ideas into digestible formats so that learners can efficiently process, apply, and retain knowledge.

When learning concepts like quadratic functions and interval calculations, visualization tools and step-by-step guidance are beneficial. These educational strategies enhance student engagement and understanding by making abstract concepts more concrete.

• Develop practical skills through exercises and real-life applications.
• Use resources like step-by-step solutions, video tutorials, and quizzes to reinforce learning.
• Foster critical thinking by encouraging problem-solving from multiple angles.

Using diverse teaching methods and focusing on different learning styles ensures all students have the opportunity to grasp essential mathematical concepts.