Problem 33 - Calculus with Concepts in Calculus

Question

Have a computer or graphics calculator plot the graph of the function on the given \(x\) window. Alter the \(y\) window as needed in order to be able to discern the major features of the graph. By reading off the \(y\) coordinate of the lowest point on the graph, approximate the smallest value assumed by the function on the given interval. It may be helpful to adjust the window. $$ f(x)=210 x^{4}-107 x^{3}+18 x^{2}-x ;[0,0.5] $$

Step-by-Step Solution

Answer

The smallest value on \([0, 0.5]\) can be approximately found by examining the graph.

Step 1: Determine the Graphing Window for x

For the function \( f(x) = 210x^4 - 107x^3 + 18x^2 - x \), we are tasked with graphing it within the interval \([0, 0.5]\). This means we need to set the x-axis of our graphing tool to show values starting from \(0\) to \(0.5\).

Step 2: Use a Graphing Tool

Enter the function \( f(x) = 210x^4 - 107x^3 + 18x^2 - x \) into a computer or a graphing calculator. Set the x-axis window to \([0, 0.5]\), as determined in Step 1. Temporarily set the y-axis window to a rough range like \([-1, 1]\) or use auto-scaling if available to get an initial image of the graph.

Step 3: Adjust the y-Axis Window

After obtaining an initial graph, you may find that fine-tuning the range of the y-axis is necessary to see the major features clearly. Adjust the y-axis window in the graphing tool incrementally until you can observe the lowest point on the graph clearly and ensure the entire curve from \(x = 0\) to \(x = 0.5\) is visible.

Step 4: Identify the Lowest Point

Inspect the graph to determine the lowest point on the segment between \(x = 0\) and \(x = 0.5\). This involves zooming in and accurately tracing or calculating the position of this minimum point.

Step 5: Approximate the Minimum y-Value

By closely examining the plotted graph, read off the \(y\)-coordinate corresponding to the lowest point. Use this approximation to state the smallest value of the function \(f(x)\) in the interval \([0, 0.5]\).

Key Concepts

Graphing Calculators

Graphing calculators are essential educational tools for visualizing mathematical functions. They allow students to input polynomial functions, set specific ranges for the x-axis and y-axis, and graphically analyze these functions. By using a graphing calculator, you can better understand the behavior and characteristics of complex equations.
One of the significant advantages is their ability to handle a variety of functions, especially those involving higher-degree polynomials, like quartic functions such as the one given in this exercise. Having access to this tool allows you to easily manipulate the graph appearance.

• First, input the polynomial function into the calculator.
• Next, adjust the graphing window to focus on the interval of interest.
• Utilize zoom and measurement tools to further enhance the graph view.

This process helps in identifying crucial graph characteristics like the function's slope, direction of curvature, intercepts, and local minima or maxima.

x-axis and y-axis adjustment

Adjusting the x-axis and y-axis is a crucial part of graphing functions accurately. For polynomial functions, these adjustments help display the graph's critical features more clearly. In our specific function, the x-axis is set from 0 to 0.5 because we are interested in this particular domain.
Start by setting a basic y-axis range, such as \([-1, 1]\), to get an approximation of the function's behavior. Initial scaling might not always display all details. Adjusting the y-axis incrementally allows you to capture finer graph details. Change the y-axis window settings gradually to reveal peak values and turning points.

This targeted adjustment makes it easier to trace the function's behavior, helping you identify points like relative minima more efficiently. As a result, this method leads to a more precise understanding of the functional behavior over the specified domain.

Minimum Value Approximation

Approximating the minimum value of a function on a given interval is a common task in calculus and analysis. In this context, it involves using the graph plotted by your calculator. Begin by visually inspecting the graph within the set window \([0, 0.5]\). This step ensures you identify the graph features, particularly the local minimum within the specified x-values.
Once the graph is clearly defined, look for the point where the y-coordinate is smallest. This point marks the minimum value on the graph. It's important to zoom in on this point to make an accurate approximation.

After identifying the lowest point on the graph, carefully read off the corresponding y-coordinate. This value is the minimum approximation for the polynomial function in the given interval. By using the graph, this step can be efficiently completed without heavy computation.

Polynomial Function Graphing

Graphing polynomial functions involves understanding the shape and behavior of the graph. Unlike linear or quadratic graphs, polynomial functions like quartic graphs have more complex behavior, showcasing multiple turning points. For the given function \(f(x) = 210x^4 - 107x^3 + 18x^2 - x\), the graph's characteristics depend on both the coefficients and the degree of the polynomial.
Higher degree polynomials, such as quartics, may have up to \(n-1\) turning points, where \(n\) is the degree of the polynomial. As observed, polynomial functions tend to exhibit curves that change direction, showing one or more peaks and troughs.

When graphing a polynomial, start by identifying general properties:

• The leading term's coefficient (in this case, 210) affects the end behavior of the polynomial.
• Turning points can provide indications of maximum and minimum values.
• Understanding the graph's overall shape helps in approximating these critical values.

By analyzing these features, you can interpret and graph the given polynomial function more accurately, leading to deeper insights into its behavior over any interval.