Problem 55 - College Algebra with Modeling and Visualization

Question

Compute the average rate of change of \(f\) from \(x_{1}\) to \(x_{2}\). Round your answer to two decimal places when appropriate. Interpret your result graphically. $$ f(x)=x^{3}-2 x, x_{1}=2, \text { and } x_{2}=4 $$

Step-by-Step Solution

Answer

The average rate of change is 26.

Step 1: Understand the Problem

We need to calculate the average rate of change of the function \( f(x) = x^3 - 2x \) from \( x_1 = 2 \) to \( x_2 = 4 \). This is similar to finding the slope of the secant line that passes through these points on the graph of \( f \).

Step 2: Formula for Average Rate of Change

The average rate of change of a function \( f(x) \) from \( x_1 \) to \( x_2 \) is given by: \[ \text{Average rate of change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1} \]

Step 3: Calculate \( f(x_1) \) and \( f(x_2) \)

Substitute \( x_1 = 2 \) into \( f(x) = x^3 - 2x \) to get \( f(2) = 2^3 - 2 \times 2 = 8 - 4 = 4 \).Substitute \( x_2 = 4 \) into \( f(x) = x^3 - 2x \) to get \( f(4) = 4^3 - 2 \times 4 = 64 - 8 = 56 \).

Step 4: Compute the Average Rate of Change

Using the formula from Step 2, substitute the values: \[ \text{Average rate of change} = \frac{f(4) - f(2)}{4 - 2} = \frac{56 - 4}{2} = \frac{52}{2} = 26 \].

Step 5: Interpret the Result Graphically

The average rate of change is 26, meaning that between \( x = 2 \) and \( x = 4 \), the function \( f(x) \) increased by 26 units for each unit increase in \( x \). Graphically, this implies the slope of the secant line through the points (2,4) and (4,56) is 26.

Key Concepts

Secant Line

A secant line helps us understand how a function changes over an interval. It is a straight line that connects two points on the graph of a function. In this context, the secant line spans from the point where the function is evaluated at \( x_1 = 2 \) to the point at \( x_2 = 4 \). To find the properties of this line, primarily its slope, we compute the average rate of change of the function over this interval.

The slope of the secant line represents how steep the line is. It is calculated by taking the difference between the values of the function at the two given points and dividing by the difference in the \( x \)-values:

• The formula used is \( \frac{f(x_2) - f(x_1)}{x_2 - x_1} \).
• In simple terms, it tells us how much \( y \) (or the function value) changes per unit change in \( x \).

In our exercise, this value was found to be 26, meaning between \( x = 2 \) and \( x = 4 \), the function value increases quite steeply.

Polynomial Functions

Polynomial functions like the one presented in this exercise, \( f(x) = x^3 - 2x \), are algebraic expressions involving variables raised to whole number powers. These functions are defined mathematically as sums of terms, each consisting of a coefficient and a variable raised to a non-negative integer exponent.

A key feature of polynomials is their smooth, continuous curves when plotted, which makes them easier to work with in terms of calculus and other branches of mathematics. In our case, this cubic polynomial:

• Has a leading term of \( x^3 \), which plays a significant role in how the function behaves as \( x \) becomes very large or very small.
• Influences the steepness and direction of the graph as shown by the large change captured by the secant line slope.

Understanding these properties helps us appreciate how polynomial behavior informs the rate of change over different intervals.

Graphical Interpretation

The graphical interpretation of the average rate of change provides valuable visual insights into the behavior of a function. When plotting the function \( f(x) = x^3 - 2x \), we see a curve that captures how the function grows and shrinks at various intervals.

The secant line, in this case, helps us visualize this average rate of change between the points \( (2,4) \) and \( (4,56) \). If you plot these points and draw the secant line through them:

• The slope of this line, 26, indicates how steeply the function is increasing over this section of the graph.
• This tells us that between these two points, for every single increment in \( x \), the value of \( f(x) \) increases by 26.

Graphically interpreting functions and their changes allow for a more intuitive understanding of mathematical concepts and the dynamic nature of polynomials.