Question

In Exercises, use a graphing utility to graph \(f, f^{\prime}\), and \(f^{\prime \prime}\) in the same viewing window. What is the relationship among the degree of \(f\) and the degrees of its successive derivatives? In general, what is the relationship among the degree of a polynomial function and the degrees of its successive derivatives? $$ f(x)=3 x^{3}-9 x $$

Short answer

The relationship between the degree of a polynomial function and the degrees of its successive derivatives is that with each derivative taken, the degree of the resulting polynomial decreases by one. The degree of our given function \(f(x)\) is 3, the degree of its first derivative \(f'(x)\) is 2, and the degree of the second derivative \(f''(x)\) is 1.

Step-by-step solution

Derive the given function

First, we need to find the first derivative, \(f'\), of the given function \(f(x) = 3x^3 - 9x\). Using the power rule for derivatives, which states that the derivative of \(x^n\) is \(n*x^{n-1}\), we find that \(f'(x) = 9x^2 - 9 = 9(x^2 - 1)\). Now, we move on to find the second derivative, \(f''\). Again, using the power rule, we derive \(f'(x)\) to obtain \(f''(x) = 18x\).

Analyze the degrees of the derivative functions

The degree of a polynomial is determined by the highest power of \(x\). In \(f(x)\), the degree is 3 (the power of \(x\) in the highest order term, \(3x^3\)). For \(f'(x)\), the degree is 2, and for \(f''(x)\), the degree is simply 1.

Formulate the relationship

As can be observed, with each subsequent derivative taken, the degree of the polynomial reduces by one. Hence, this indicates that the relationship between the degree of a polynomial function and the degree of its successive derivatives decreases by 1 each time a derivative is taken. In general, for a polynomial of degree \(n\), its first derivative will be of degree \(n-1\), the second derivative will be of degree \(n-2\), and so on, until \(n-n\), which yields a derivative of degree 0 (a constant function).

Key concepts

Understanding the Use of Graphing Utilities

Graphing utilities are essential for students and educators to visualize mathematical concepts effectively. In the context of polynomial functions and their derivatives, graphing utilities are invaluable tools. They allow you to observe the behavior of the function and its derivatives in a single viewing window, offering immediate visual feedback on how changes in the function translate to changes in its derivatives.

For example, graphing the function \(f(x)=3x^3-9x\) and its derivatives \(f'(x)\) and \(f''(x)\) on the same set of axes helps illustrate the relationship among their degrees. The original function, due to its cubic nature, will present with a characteristic 'S' shaped curve. In contrast, the first derivative, being quadratic, will display a parabolic shape and the second derivative, a linear one. By using a graphing utility, students can clearly see how the curve of the first derivative relates to the slope of the original function, and how the second derivative graph represents the curvature of the original function.

When using such tools, it's critical to select appropriate scale settings so the relationships are clear. If scales are set too narrowly or too widely, important features of the graphs, such as intercepts and extremas, might not be adequately represented. Always check that the window settings are large enough to include all relevant behavior of the graph.

Successive Derivatives and their Degrees

When studying calculus, you will find that taking successive derivatives is a common practice to reveal various aspects of a function's behavior. Successive derivatives refer to the series of derivatives taken one after the other. Starting with the original function, known as the 'zeroth' derivative, each round of differentiation yields the next higher-order derivative.

As seen in the example with \(f(x)=3x^3-9x\), with each derivative taken, we noticed a drop in the polynomial degree by one. This pattern is consistent for all polynomial functions. The first derivative represents the rate of change of the function, the second derivative provides information about the concavity and points of inflection, and so on.

This property of derivatives to 'reduce' the degree by one each time has significant implications. It means that for any polynomial function of degree \(n\), it is guaranteed that after taking \(n\) derivatives, you will be left with a constant function (a zero-degree polynomial). Teachers and textbooks often employ this fact when teaching optimization and curve sketching since it simplifies the process of finding maxima, minima, and inflection points.

Applying the Power Rule for Derivatives

The power rule is a fundamental concept in calculus used to take the derivative of polynomial functions easily. It states that if you have a term in the form of \(x^n\), where \(n\) is any real number, its derivative is \(nx^{n-1}\). This rule helps you quickly find the derivatives without reliance on the definition of a derivative or more complex rules.

Examining the step-by-step solution of the function \(f(x)=3x^3-9x\), we applied the power rule and found that \(f'(x) = 9x^2 - 9\), and subsequently, \(f''(x) = 18x\). The power rule allows for direct computation of these derivatives without complicated calculations.

Students should practice using the power rule in different contexts to become proficient. Being able to differentiate quickly and accurately is essential for more advanced calculus topics, such as studying the behavior of functions, integration, and solving differential equations. Remember that the power rule does not apply to all functions—you cannot use it on functions that aren't polynomials. Furthermore, for negative and fractional powers of \(x\), the rule still applies but requires careful handling to prevent errors.