Problem 52 - Calculus with Concepts in Calculus

Question

In order to solve an inequality on a graphics calculator, we can graph a corresponding function and determine where it is positive and where it is negative. In Exercises \(51-52\) use the zoom feature of a graphics calculator to find an approximate solution of the inequality. Zoom until successive values of the \(x\) coordinate have identical first three digits. $$ 4 x^{3}+4 x<3 $$

Step-by-Step Solution

Answer

The approximate solution is where \(-0.9 < x < 0\).

Step 1: Rewrite the Inequality

Start by rewriting the inequality as an equation which can be graphed. The inequality is \(4x^3 + 4x < 3\). Subtract 3 from both sides to set the inequality to zero: \(4x^3 + 4x - 3 < 0\). This represents the function \(f(x) = 4x^3 + 4x - 3\) which we will graph.

Step 2: Enter the Function into the Calculator

Enter the function \(f(x) = 4x^3 + 4x - 3\) in the 'Y=' section of the graphics calculator as this will allow you to graph it.

Step 3: Graph the Function

Select the graph button on your graphics calculator to view the graph of \(f(x) = 4x^3 + 4x - 3\). You should see a cubic curve on the display.

Step 4: Use the Zoom Feature

Utilize the zoom feature of the calculator to closely inspect the area where the function crosses the x-axis, as this indicates where the function changes from positive to negative, or vice versa. Continue zooming in until the x-coordinates of the intercept have identical first three digits.

Step 5: Determine the Sign of the Function

Analyze the graph to determine the x-values where the graph is below the x-axis (negative). Since we are solving \(4x^3 + 4x - 3 < 0\), we are interested in the intervals where the function is below zero.

Key Concepts

Understanding a Graphing Calculator

A graphing calculator is a powerful tool, especially when dealing with complex functions like inequalities. It allows you to visually interpret data by plotting mathematical functions, making it easier to understand where certain conditions hold true. When you need to determine where an inequality holds, such as in this exercise, a graphing calculator displays the function as a curve. This visual representation includes:

• Points on the curve where the function equals zero.
• Intervals where the function is above or below the x-axis.
• Function's behavior through different regions of the x-axis.

Graphing calculators provide multiple functionalities such as calculating roots, intersections, and optimizing values. By entering the function into the calculator, you get an immediate visual on the screen. This is crucial, especially for complex functions, as you can instantly see where the function changes signs or behavior.

Using the Zoom Feature Effectively

The zoom feature on a graphing calculator is essential when trying to pinpoint specific values or occurrences on a graph. When dealing with inequalities and similar tasks, you often need to zoom in to get more detailed information about where exactly the curve intersects the x-axis. This process involves:

• Narrowing down the view to focus on critical points like intercepts.
• Ensuring that calculations or visual assessments are as accurate as possible.
• Continuing to zoom until successive x-values show consistency in the significant digits required by your task.

For instance, in the given inequality, by zooming in, you look for x-values where they have identical first three digits. This precision helps prevent errors and provides a clear range where the inequality criteria are met.

Cubic Functions and Their Properties

Cubic functions are polynomial functions of the form \(f(x) = ax^3 + bx^2 + cx + d\), where the highest degree is three. These functions often have more complex behaviors than linear or quadratic functions, including:

• One or more turning points.
• A possible inflection point where the curvature changes.
• One real root and possibly two complex roots, or three real roots.

In our exercise, the cubic function is \(f(x) = 4x^3 + 4x - 3\). The cubic nature of this function means the graph can cross the x-axis up to three times, indicating up to three changes in sign. This is why visual representation on a graphing calculator is significant—it allows you to see where these intersections occur. Identifying the intervals where the function is negative (below the x-axis) helps solve inequalities like the one in this exercise.