Problem 19 - Calculus Early Transcendentals: Pearson New International Edition

Question

Without doing any calculating, find each derivative. (a) \(D_{x}^{4}\left(3 x^{3}+2 x-19\right)\) (b) \(D_{x}^{12}\left(100 x^{11}-79 x^{10}\right)\) (c) \(D_{x}^{11}\left(x^{2}-3\right)^{5}\)

Step-by-Step Solution

Answer

(a) 0; (b) 0; (c) 0.

Step 1: Analyze the Function for Part (a)

For part (a), we have the function \(3x^3 + 2x - 19\). Notice that this is a third-degree polynomial. Since we are taking the fourth derivative \(D_{x}^{4}\), which is one order higher than the polynomial's degree, the derivative will be zero. Higher derivatives of polynomials become zero when the order exceeds the polynomial's degree.

Step 2: Analyze the Function for Part (b)

In part (b), the function is \(100 x^{11} - 79 x^{10}\). It's an expression with the highest degree of 11. When taking the 12th derivative \(D_{x}^{12}\), we exceed the degree of even the highest polynomial term. As such, the 12th derivative is zero.

Step 3: Analyze the Function for Part (c)

In part (c), the function is \((x^2 - 3)^5\), which represents a composition of functions. To apply the 11th derivative \(D_{x}^{11}\), we would need to expand this expression and find terms of degree 11 or higher for the derivative to be non-zero. However, the original expression's maximum possible degree is less than 11, meaning all terms vanish by the 11th derivative.

Key Concepts

Higher Order Derivatives

In calculus, derivatives measure how a function changes as its input changes. Higher order derivatives extend this concept. They are simply derivatives of derivatives.
For instance, if you take the first derivative of a function, that tells you the rate of change. The second derivative provides information about how that rate of change is itself changing. As you continue to find derivatives, you look further into these changes.
When working with polynomials, higher order derivatives have particular behavior:

• If the degree of a polynomial is less than the order of the derivative being taken, the result is zero. This is because each differentiation reduces the degree of the polynomial until it can't be reduced any further.
• Language like \(D_{x}^{n}\) is used, where \(n\) specifies which derivative is being sought.

This concept was crucial for the exercise, where the derivatives of the polynomial expressions were calculated and shown to vanish after a certain point.

Polynomial Degree

The degree of a polynomial indicate the highest power of the variable in the expression. For example, in the expression \(3x^3 + 2x - 19\), the degree is 3 because of the \(x^3\) term.
Understanding the degree is essential when finding derivatives because:

• The degree of the polynomial dictates the number of times it can be differentiated before it reaches zero.
• If you try to find a derivative of order higher than the degree, you won't have any terms left. Each differentiation reduces the degree by one.

The exercise illustrates this by showing that, when taking derivatives higher than the polynomial degree of our function, the result is zero. This is why part (a) resulted in zero when a fourth derivative was taken from a third-degree polynomial.

Function Analysis

Function analysis involves breaking down and understanding the behavior and structure of a function.
When analyzing a function for derivatives, especially higher order ones, you look at:

• The composition of the function: Is it a simple polynomial? Is it a combination of different functions, like powers or products?
• The degree of the highest terms and how this degree affects the derivative process.

For example, in part (c) of the original exercise, the function \((x^2-3)^5\) is a more complex structure. Instead of a straightforward polynomial, it's a power of a polynomial. Analyzing it requires realizing that expanding it could give you a degree larger than 2, but never as large as 10 or more, making the eleventh derivative result zero.