Problem 13 - Algebra and Trigonometry

Question

A function is given. Determine the average rate of change of the function between the given values of the variable. $$ f(x)=x^{3}-4 x^{2} ; \quad x=0, x=10 $$

Step-by-Step Solution

Answer

The average rate of change of the function is 60.

Step 1: Identify the given function and interval

The given function is \( f(x) = x^3 - 4x^2 \). We need to find the average rate of change between \( x=0 \) and \( x=10 \). This average rate of change can be calculated using the formula for the difference quotient, which is \( \frac{f(b) - f(a)}{b - a} \), where \( a \) and \( b \) are the endpoints \( x=0 \) and \( x=10 \), respectively.

Step 2: Evaluate the function at the endpoints

First, we evaluate the function at the starting point \( x=0 \): \[ f(0) = 0^3 - 4(0)^2 = 0. \] Next, evaluate the function at the endpoint \( x=10 \): \[ f(10) = 10^3 - 4(10)^2 = 1000 - 400 = 600. \]

Step 3: Apply the difference quotient formula

Using the difference quotient formula, calculate the average rate of change: \[ \frac{f(10) - f(0)}{10 - 0} = \frac{600 - 0}{10} = \frac{600}{10} = 60. \]

Key Concepts

Understanding Algebra

Algebra is a branch of mathematics that uses symbols and letters to represent numbers and quantities in mathematical expressions and equations. It helps us solve problems that involve unknown values. In algebra, we often work with variables, which are symbols (like $x$) that stand for values that can change.

• **Expressions:** These are combinations of numbers, variables, and operations. For example, $x^3 - 4x^2$ is an expression found in our given function.
• **Equations:** Equations are like expressions set equal to something, which allows us to solve for unknowns. Here, the function $f(x) = x^3 - 4x^2$ creates values for different values of $x$.

To find the average rate of change of a function between two points, we apply algebraic techniques like simplifying expressions and substituting values into formulas. This step-by-step approach ensures the problem is broken down into manageable parts.

Difference Quotient Method

The difference quotient is a fundamental concept in calculus but is also used in algebra to find the average rate of change of a function. It provides us with the slope of the secant line that connects two points on a curve. The formula is:
\[\frac{f(b) - f(a)}{b - a}\]where:

• \(f(b)\) is the function value at the endpoint \(b\).
• \(f(a)\) is the function value at the endpoint \(a\).
• \(b - a\) is the difference in the \(x\) values or the interval length.

For the function \(f(x) = x^3 - 4x^2\) over the interval \(x=0\) to \(x=10\), the difference quotient helps compute the average rate of change. It tells us how much \(f(x)\) changes, on average, between these two \(x\) values.

Functions in Mathematics

Functions are like machines in mathematics that take an input, do something to it, and give an output. They are essential in understanding relationships between variables. In our exercise, we deal with the function $f(x) = x^3 - 4x^2$.

• **Input and Output:** We input a value (like $x = 0$ or $x = 10$) into the function to find the output (evaluated as $f(0) = 0$ and $f(10) = 600$).
• **Continuous Change:** Functions allow us to study how the output changes continuously as the input changes. This is crucial in finding rates of change, like our average rate of change from $f(0)$ to $f(10)$.

Understanding functions helps us grasp how equations describe real-world phenomena, making them a powerful tool for anyone studying math or sciences.