Problem 58 - Thomas Calculus in SI Units

Question

a. Construct a polynomial \(f(x)\) that has zeros at \(x=-2,-1,0\) 1, and 2. b. Graph \(f\) and its derivative \(f^{\prime}\) together. How is what you see related to Rolle's Theorem? c. Do \(g(x)=\sin x\) and its derivative \(g^{\prime}\) illustrate the same phenomenon as \(f\) and \(f^{\prime} ?\)

Step-by-Step Solution

Answer

Polynomial: \( f(x) = x^5 - 5x^3 + 4x \). Rolle's theorem applies to both \( f, f' \) and \( g, g' \) as each derivative has zero between their roots.

Step 1: Construct the Polynomial

To construct a polynomial \( f(x) \) with zeros at \( x = -2, -1, 0, 1, \) and \( 2 \), we form a polynomial in its factored form. The polynomial can be written as \( f(x) = (x + 2)(x + 1)x(x - 1)(x - 2) \) because each zero, or root, appears as a factor.

Step 2: Write the Polynomial in Expanded Form

Expand the factors to write the polynomial as a standard expression. Start by multiplying the pairs: \( (x + 2)(x - 2) = x^2 - 4 \) and \( (x + 1)(x - 1) = x^2 - 1 \). Now, \( f(x) = (x^2 - 4)(x^2 - 1)x \). Expand: \( (x^2 - 4)(x^2 - 1) = x^4 - 5x^2 + 4 \). Hence, \( f(x) = x(x^4 - 5x^2 + 4) = x^5 - 5x^3 + 4x \).

Step 3: Find Derivative of the Polynomial

Differentiate the polynomial \( f(x) = x^5 - 5x^3 + 4x \) using the power rule. The derivative is \( f'(x) = 5x^4 - 15x^2 + 4 \).

Step 4: Graph the Polynomial and Its Derivative

Using a graphing tool, plot both \( f(x) = x^5 - 5x^3 + 4x \) and \( f'(x) = 5x^4 - 15x^2 + 4 \). Notice the points where the graph of \( f(x) \) touches or crosses the x-axis (the zeros) and observe the graph of \( f'(x) \) at those points.

Step 5: Relate to Rolle's Theorem

Rolle's Theorem states that if a function is continuous on \([a, b]\), differentiable on \((a, b)\), and \( f(a) = f(b) = 0 \), there is at least one \( c \) in \( (a, b) \) where \( f'(c) = 0 \). In our graph, between consecutive zeros of \( f \), \( f'(x) \) has zeros, illustrating this theorem.

Step 6: Analyze \( g(x) = \sin x \) and Its Derivative

\( g(x) = \sin x \) is a periodic function with zero-crossings at \( x = n\pi \) for \( n \in \mathbb{Z} \). Its derivative, \( g'(x) = \cos x \), is zero where \( \sin x \) has local extrema. Therefore, like \( f(x) \), \( g(x) \) satisfies the conditions of Rolle's Theorem between its zeros.

Key Concepts

Polynomial Construction

To construct a polynomial with specific zeros is like piecing together a puzzle. A zero of a polynomial is a value of \( x \) such that the polynomial equals zero. If \( x = -2, -1, 0, 1, 2 \) are zeros, each zero corresponds to a factor. Thus, the polynomial is found in its factored form: \( f(x) = (x + 2)(x + 1)x(x - 1)(x - 2) \). Each factor \((x - a)\) represents a zero \( a \).

To express it in standard form, simply expand the factors. Begin by pairing the factors:

• \((x + 2)(x - 2) = x^2 - 4\)
• \((x + 1)(x - 1) = x^2 - 1\)

Combine them with the remaining \(x\): \( f(x) = (x^2 - 4)(x^2 - 1)x \). Expand to get the final polynomial:

\( f(x) = x^5 - 5x^3 + 4x \). By breaking down the construction, you see how zeros connect to polynomial form.

Graphing Functions

Graphing a function involves plotting its curve on a plane, providing a visual display of its behavior. For a polynomial like \( f(x) = x^5 - 5x^3 + 4x \), a graph reveals where it crosses the x-axis, these being its zeros. To better understand its dynamics, graph both the function and its derivative \( f'(x) = 5x^4 - 15x^2 + 4 \) on the same plane.

Observing this:

• Notice where \( f(x) \) crosses or touches the x-axis.
• Check the corresponding points on \( f'(x) \), especially where it becomes zero or changes sign.

These points often mark local extrema, confirming the presence of zeros of the derivative due to changes in the original function. Graphing highlights key behaviors and intersections, critical for analyzing the impacts of Rolle's Theorem.

Differentiation

Differentiation is the process of finding a function's derivative. This involves determining the rate at which the function changes. For the polynomial \( f(x) = x^5 - 5x^3 + 4x \), apply the power rule of differentiation to find \( f'(x) \).

Use the power rule: If \( f(x) = ax^n \), then \( f'(x) = n \, ax^{n-1} \). Applying this, we get:

• \( rac{d}{dx}(x^5) = 5x^4 \)
• \( rac{d}{dx}(-5x^3) = -15x^2 \)
• \( rac{d}{dx}(4x) = 4 \)

Thus, \( f'(x) = 5x^4 - 15x^2 + 4 \). Differentiation reveals changes in slope, leading to understanding of critical points and behaviors like those seen in Rolle's Theorem.

Sine Function and Its Derivative

The sine function, \( g(x) = \sin x \), illustrates periodic behavior, providing an excellent example of differentiation in circular functions. The derivative, \( g'(x) = \cos x \), offers insight into changes in \( g(x) \).

For \( \sin x \):

• Zero-crossings occur at \( x = n\pi \) where \( n \in \mathbb{Z} \).
• The function reaches peaks and troughs halfway between these zeros.

Similarly:

• \( g'(x) = \cos x \) is zero when \( g(x) \) is at these extrema.

Between the zeros of sine, the derivative is zero, creating a parallel with the polynomial and demonstrating Rolle's Theorem. The nature of the sine function makes it a natural example of periodicity coupled with its derivative.