Problem 10 - Thomas Calculus

Question

Find an appropriate graphing software viewing window for the given function and use it to display its graph. The window should give a picture of the overall behavior of the function. There is more than one choice, but incorrect choices can miss important aspects of the function. \(\begin{equation} f(x)=x^{2}\left(6-x^{3}\right) \end{equation}\)

Step-by-Step Solution

Answer

A suitable window is x from -3 to 3 and y from -50 to 50.

Step 1: Identify key features of the function

We start by analyzing the function to determine any key features. The function is given by \(f(x) = x^2(6-x^3)\). It is a polynomial of degree 5, as the highest power of \(x\) in the expression is \(x^5\). We expect the graph to resemble a flattened 'S' shape typical of degree 5 polynomials, with changes in direction.

Step 2: Determine intercepts and symmetry

Determine where the function intersects the axes. For the y-intercept, substitute \(x = 0\), giving \(f(0) = 0\). For x-intercepts, set \(f(x) = 0\): \(x^2(6-x^3) = 0\) solves as \(x = 0\) or \(x^3 = 6\), giving \(x = \sqrt[3]{6}\). The function is even due to the \(x^2\) factor but not symmetric about the y-axis since it also includes \(-x^3\).

Step 3: Identify potential extremes and inflection points

Find critical points by taking the derivative \(f'(x)\). The derivative is \(f'(x) = 2x(6-x^3) - 3x^5\). Set this derivative to zero to find critical points. Solve \(2x(6-x^3) = 3x^5\) or factor as \(x(12 - 5x^3) = 0\). This gives \(x = 0\) or \(x = \sqrt[3]{\frac{12}{5}}\). These values should be tested to check if they indeed are critical points.

Step 4: Choose an appropriate graphing window

Considering the intercepts and critical points along with the polynomial degree, set a window that captures these features. A reasonable window might be from \(-3\) to \(3\) on the x-axis to capture the function's extrema and intercepts and \(-50\) to \(50\) on the y-axis. This window ensures the key features and behavior are visible.

Step 5: Graph the function

Using the determined window, graph the function with graphing software. Confirm that the chosen window effectively displays critical elements of the graph, showing how it behaves overall. Ensure that it includes the intercepts and key direction changes identified.

Key Concepts

Function Intercepts

When studying functions, particularly polynomial ones, intercepts are crucial. They are the points where the graph crosses the axes. This gives insights into the function's behavior. For any function \( f(x) \), finding intercepts is like uncovering its basic DNA. Let's break this down:

• Y-Intercept: This is the point where the function crosses the y-axis. We find it by setting \(x = 0\) in the function equation. Here, \( f(0) = x^2(6-x^3) \) simplifies directly to 0. So, the y-axis intercept is at (0,0).
  
• X-Intercepts: These are the roots of the function, where it crosses the x-axis. We determine them by solving \( f(x) = 0 \). In this problem, \( x^2(6-x^3) = 0 \) gives us two intercepts: \( x = 0 \) and \( x = \sqrt[3]{6} \).
  

Understanding intercepts helps in predicting how the graph behaves as it approaches and moves through these points on the axes.

Critical Points

Critical points are significant in graphing as they indicate where the function's slope changes, identifying potential maxima, minima, or points of inflection. To find these, we simply take the function's derivative and set it to zero.
First off, calculating the derivative provides insight into the function's rate of change. For our function, \( f(x) = x^2(6-x^3) \), the derivative is \( f'(x) = 2x(6-x^3) - 3x^5 \). Simplifying this, we have \( x(12 - 5x^3) = 0 \).
From solving the equation \( x(12 - 5x^3) = 0 \), we find it factors to \( x = 0 \) and \( x = \sqrt[3]{\frac{12}{5}} \). These are the x-values of critical points. They are not just numbers—they guide us on where significant changes occur on the graph.
To finalize whether these points are peaks, troughs, or simply directional changes, one would typically calculate the second derivative or examine the sign change of \( f'(x) \) around these points.

Graphing Software

Graphing software is invaluable for visualizing complex polynomial equations like \( f(x) = x^2(6-x^3) \). It helps communicate the function's story instantly, showing where the intercepts and critical points align in a visually engaging way.
Why use graphing software? Here are some benefits:

• Visual Clarity: Many polynomial functions have intricate behaviors. Changes in direction, like those in our degree-5 function, are more easily understood when seen visually.
  
• Flexibility: You can tweak the viewing window to include all important features like intercepts and critical points. For our example, a window from \([-3, 3]\) along the x-axis and \([-50, 50]\) on the y-axis gives a nice holistic view.
  
• Efficiency: Graphing by hand takes time and can be prone to error, especially with complex functions. Software handles these calculations quickly, allowing more focus on analysis.

In summary, graphing software doesn't merely draw the function; it serves as a canvas that illustrates key mathematical insights, helping deepen our understanding of the function's behavior.