Problem 10 - Thomas Calculus

Question

In Exercises 5–30, determine an appropriate viewing window for the given function and use it to display its graph. $$ f(x)=x^{2}\left(6-x^{3}\right) $$

Step-by-Step Solution

Answer

Use a viewing window of \( x: [-3, 3] \) and \( y: [-100, 100] \) to view the function.

Step 1: Analyze the function

The function given is \( f(x) = x^2(6 - x^3) \). It is a polynomial function, which means it is continuous and smooth. The degree of the polynomial (the highest exponent of \(x\)) is 5, because when expanded, \(x^2\) multiplied by \(-x^3\) gives \(-x^5\). So, this function will have up to 5 roots and turns.

Step 2: Determine the intercepts

First, find the x-intercepts by setting \( f(x) = 0 \). This gives \( x^2(6 - x^3) = 0 \). The solutions are \( x = 0 \) and \( 6 - x^3 = 0 \). Solving \( 6 - x^3 = 0 \) yields \( x^3 = 6 \), so \( x = \sqrt[3]{6} \approx 1.82 \). Thus, the x-intercepts are at \( x = 0 \) and \( x \approx 1.82 \). For the y-intercept, evaluate \( f(0) \), which is 0. Hence, the y-intercept is also 0.

Step 3: Investigate the end behavior

As \( x \to \infty \), the dominant term is \(-x^5\), which goes to \(-\infty\) because the power is odd and the coefficient is negative. As \( x \to -\infty \), \( -x^5 \) goes to \( \infty \). This tells us the graph will fall to the right and rise to the left.

Step 4: Choose an initial viewing window

Using the intercepts and the end behavior observation, start with a viewing window. Consider \([-3, 3]\) for the x-axis to include the roots and some behavior changes. For the y-axis, start with \([-50, 50]\) to capture potential extreme values in \(f(x)\).

Step 5: Adjust if needed

After graphing with this initial window, check if all intercepts and critical points are clear. If turns or extreme values are not visible, expand the y-axis to \([-100, 100]\). Adjust further if necessary to capture essential features of the graph.

Key Concepts

Intercepts

Intercepts are important points on a graph where the function crosses an axis. They provide key insights about the polynomial function. For a function like \( f(x) = x^2(6-x^3) \), finding intercepts helps in identifying where the curve will touch or cross the axis.

### X-InterceptsTo find the x-intercepts, set the function equal to zero: \( x^2(6-x^3) = 0 \). Solving this, you get two critical points:

• \( x = 0 \), because any number squared that equals zero must be zero.
• \( 6 - x^3 = 0 \) leads to \( x^3 = 6 \), hence \( x \approx 1.82 \).

These points tell us that the graph will cross the x-axis at these values.

### Y-InterceptTo find the y-intercept, evaluate the function at \( x = 0 \): \( f(0) = 0 \). Therefore, the y-intercept is at the origin (0,0). So, the graph also crosses the y-axis here. Understanding these intercepts is crucial for sketching an accurate graph as they guide where the curve intersects the axes.

End Behavior

The end behavior of a polynomial function describes how it behaves as \( x \) moves towards infinity or negative infinity. For the function \( f(x) = x^2(6 - x^3) \), we consider the highest degree term to predict the end behavior.

### Positive and Negative Infinity BehaviorThe highest degree of the polynomial determines the end behavior. Here, after expanding, the term \(-x^5\) is dominant. This is because 5 is the highest exponent of \( x \).

• As \( x \to \infty \) (x goes to positive infinity), the term \(-x^5\) tends to \(-\infty\). This means the graph falls on the right.
• As \( x \to -\infty \) (x goes to negative infinity), the term \(-x^5\) tends to \(\infty\). Thus, the graph rises on the left.

These behaviors are essential for predicting the overall shape of the graph. Knowing that the curve descends to the right and ascends to the left helps in planning the viewing window and understanding the polynomial's nature.

Viewing Window Selection

Choosing an appropriate viewing window is crucial for accurately visualizing a polynomial function. It ensures that all critical features of the graph are visible.

### Selecting the X-Axis and Y-Axis RangesBased on intercepts and end behavior, a starting point for plotting \( f(x) = x^2(6-x^3) \) could be:

• X-Axis: Setting the range between \([-3, 3]\) captures key intercepts and potential behavior changes.
• Y-Axis: Starting from \([-50, 50]\) helps include possible extreme values of \( f(x) \).

These are initial values that might need adjustment. After examining the graph, you may find it necessary to change the y-axis, perhaps expanding to \([-100, 100]\) if needed, to make all features clear.

### Importance of Window AdjustmentAdjustments ensure that both intercepts and turning points are clear. Without proper adjustment, crucial details can be missed. Hence, experimenting with different window sizes is often necessary to capture the most informative graph.

Degree of Polynomial

The degree of a polynomial is the highest power of the variable \( x \) in the function. It provides valuable information about the polynomial's features such as number of roots and general shape.

### Determining the DegreeIn the function \( f(x) = x^2(6-x^3) \), we can expand it as \( x^2 \cdot (6-x^3) = 6x^2 - x^5 \). Here, the degree is 5 because the term \(-x^5\) has the highest power.

### Implications of the Degree

• A polynomial of degree 5 can have up to 5 roots or zero crossings.
• It can also have up to \( 4 \) turns—which are the points where the direction of the curve changes.
• The end behavior of the polynomial can be inferred from the degree, as seen with the highest-term analysis of \(-x^5\).

Understanding this concept helps predict the complexity of a graph and the number of features like intercepts and turns, you might expect to observe. It's a critical first step in analyzing any polynomial function.