Problem 21 - Algebra and Trigonometry

Question

\(17-28\) 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 is 60.

Step 1: Understanding the Problem

The average rate of change of a function between two values can be thought of as the slope of the straight line connecting the points on the graph of the function at these two values. Mathematically, it's the change in the function values divided by the change in the input values.

Step 2: Calculate Function Values at Given Points

Calculate the values of the function at the given points: - For \(x = 0\), \(f(0) = 0^3 - 4(0)^2 = 0\).- For \(x = 10\), \(f(10) = 10^3 - 4(10)^2 = 1000 - 400 = 600\).

Step 3: Find the Change in Function Values

Determine the difference between the function values at the given points. So, the change in \(f(x)\) is \(f(10) - f(0) = 600 - 0 = 600\).

Step 4: Calculate the Change in Input Values

Find the change in the input values, which is given by the difference \(x_2 - x_1 = 10 - 0 = 10\).

Step 5: Compute the Average Rate of Change

The average rate of change of the function on the interval \([0, 10]\) is calculated using the formula:\[\text{Average Rate of Change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1} = \frac{600 - 0}{10}\]This simplifies to \(60\).

Key Concepts

Function Evaluation

Function evaluation involves assessing a function at specific input values. When given a mathematical function, such as \( f(x) = x^3 - 4x^2 \), you substitute the input variable with specific numbers. For instance, to find \( f(0) \), replace \( x \) with 0, yielding \( f(0) = 0^3 - 4(0)^2 = 0 \). Similarly, for \( x = 10 \), replace \( x \) with 10 to get \( f(10) = 10^3 - 4(10)^2 = 1000 - 400 = 600 \). Evaluating functions allows you to determine the output or value of the function at particular points. This skill is key in understanding changes in function behavior over different domains.

Slope of a Line

The slope of a line measures the steepness or incline between two points on a graph. In the context of a function, this is often computed as the "average rate of change." Imagine connecting two points on a graph of a function with a straight line. The slope of this line represents how much the function's output changes per unit increase in input. It is calculated as the difference in the function's value at the two points \((f(x_2) - f(x_1))\) divided by the difference in the corresponding input values \((x_2 - x_1)\). For the function \( f(x) = x^3 - 4x^2 \) between \( x = 0 \) and \( x = 10 \), the slope is \( 60 \), meaning the function increases by 60 units for every 1 unit increase in \( x \) over the interval.

Interval Calculation

Interval calculation involves determining the difference between two values on an axis. This is crucial for computing the average rate of change. Consider a mathematical function defined over an interval \([x_1, x_2]\). For instance, with \( x_1 = 0 \) and \( x_2 = 10 \), the interval calculation \( x_2 - x_1 \) equals 10. Computing intervals helps us understand the range over which changes are measured, such as how much the input (\( x \) in a function) is allowed to vary. This calculation is also integral in determining the slope or rate of change in mathematical problems.

Polynomial Functions

Polynomial functions are mathematical expressions involving sums of powers of variables. They have terms like \( x^3 \) or \( -4x^2 \), each possibly with a coefficient. The function \( f(x) = x^3 - 4x^2 \) is an example of a polynomial function, where the highest power of \( x \) is the degree of the polynomial. Polynomials are important due to their numerous applications, including modeling real-world phenomena and solving equations. Understanding polynomial functions involves evaluating them at various points, analyzing their degree, and interpreting their graph's shape. These functions are pivotal in learning more complex algebraic and calculus concepts.