Problem 47 - College Algebra and Calculus: An Applied Approach

Question

In Exercises, use a graphing utility to graph \(f_{t} f^{\prime}\), 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? $$ f(x)=x^{2}-6 x+6 $$

Step-by-Step Solution

Answer

The degree of the \(k^{th}\) derivative of a polynomial of degree \(n\) is \(n-k\), for \(k\leq n\) and is 0 for \(k>n\).

Step 1: Calculate the first derivative of \(f\)

The derivative of a function can be calculated using the power rule which states that: the derivative of \(x^n\) is \(n*x^{n-1}\). The first derivative of the given function, \(f(x)=x^2-6x+6\) is:\[f'(x)=\frac{d}{dx}x^{2}-\frac{d}{dx}6x+\frac{d}{dx}6\] After applying the power rule, we get that:\[f'(x)= 2x-6\]

Step 2: Calculate the second derivative of \(f\)

Similarly, the second derivative of \(f\) can be calculated by differentiating the first derivative. So,\[f''(x)=\frac{d}{dx}(2x-6)\]and after calculating the above derivative, we get that:\[f''(x)= 2.\]

Step 3: Analyze the degrees of the function and its derivatives

The degree of a polynomial is the highest power of \(x\) in its expression. The original function \(f(x)= x^2-6x+6\) is of degree 2, its first derivative \(f'(x)= 2x-6\) is of degree 1 and the second derivative \(f''(x)=2\) is of degree 0. As we can see, with each successive derivative, the degree of the polynomial decreases by 1.

Step 4: Provide a general conclusion

The relationship between the degree of a polynomial and the degrees of its successive derivatives is as follows: With each successive derivative, the degree of the polynomial decreases by 1. In other words, the degree of the \(k^{th}\) derivative of a polynomial of degree \(n\) is \(n-k\), for \(k\leq n\). For \(k>n\), the derivative is 0.

Key Concepts

Degree of Polynomial

When we talk about polynomials, the 'degree' is an important concept that indicates the highest power of the variable in the expression. For example, in the polynomial \(f(x) = x^2 - 6x + 6\), the highest exponent on \(x\) is 2. This means the degree of this polynomial is 2.

Degrees of polynomials guide us in understanding how the function behaves as the input values grow large or small. When you differentiate a polynomial, you reduce its degree by one for each differentiation. This is because the derivative of a term \(x^n\) becomes \(nx^{n-1}\).

Understanding the degree of a polynomial helps in quickly assessing the complexity and behavior of the function and its graph.

• The degree of the original polynomial is 2.
• The degree of the first derivative, \(f'(x) = 2x - 6\), is 1.
• The second derivative, \(f''(x) = 2\), is a constant, so its degree is 0.

Graphing Polynomial Functions

Graphing polynomial functions helps visualize the roots, turning points, and overall behavior. A graph gives insights into crucial features, such as where the function crosses the x-axis, known as the roots or zeros.

For the polynomial \(f(x) = x^2 - 6x + 6\), the graph is a parabola opening upwards because the coefficient of \(x^2\) is positive. Graphing both \(f(x)\) and its derivatives can reveal the relationship between these different forms. The first derivative, \(f'(x) = 2x - 6\), indicates the slope or steepness at any point on the graph of \(f(x)\).

Graphing utilities can show how polynomials and their derivatives change

• Where the graph levels off or its turning points occur, is where the first derivative equals zero.
• The second derivative \(f''(x) = 2\) shows constant concavity, which means the original graph is always curving upwards.

Successive Derivatives

Successive derivatives involve differentiating a function multiple times. This reveals additional properties of the original function, such as its concavity and inflection points.

Consider the function \(f(x) = x^2 - 6x + 6\): its successive derivatives provide important clues:

• The first derivative \(f'(x) = 2x - 6\) tells us about the function's rate of change. It reveals where the graph is increasing or decreasing.
• The second derivative \(f''(x) = 2\) provides information on the concavity. Because it is constant, the graph of \(f(x)\) is always concave up.

As we find more derivatives, if possible, the degree continues to decrease by one each time. Eventually, successive derivatives of a polynomial will reach a degree of zero, beyond which they become zero, indicating no further changes in the function's behavior.