Problem 60 - University Calculus: Early Transcendentals

Question

Rolle's Theorem 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

Rolle's Theorem applies to both the polynomial and sine functions, showing critical points between zeros.

Step 1: Construct the Polynomial Function

To construct a polynomial function that has zeros at \(x=-2, -1, 0, 1,\) and \(2\), we can write the polynomial in factored form using these zeros. The function is:\[ f(x) = (x+2)(x+1)x(x-1)(x-2) \]Expanding this product will give us the polynomial in standard form, but it's not necessary for our analysis of Rolle's Theorem.

Step 2: Understand Rolle's Theorem

Rolle's Theorem states that if a function \(f(x)\) is continuous on \([a, b]\), differentiable on \((a, b)\), and \(f(a) = f(b)\), then there exists at least one \(c\) in \((a, b)\) such that \(f'(c) = 0\). Our polynomial satisfies these criteria between any two consecutive zeros.

Step 3: Graph the Function and its Derivative

Graph the polynomial \(f(x) = (x+2)(x+1)x(x-1)(x-2)\) and its derivative. The polynomial graph crosses the x-axis at \(-2, -1, 0, 1,\) and \(2\). Between each pair of consecutive roots, Rolle's Theorem ensures \(f'(x) = 0\) at least once, meaning there are critical points where the slope of the tangent is zero.

Step 4: Analyze the Graph of Sine and its Derivative

Consider \(g(x) = \sin(x)\). Its derivative is \(g'(x) = \cos(x)\). Between any two points where the sine function is zero (e.g., \(x=0\) and \(x=\pi\)), the cosine function is also zero, meeting the conditions of Rolle's Theorem. Just like the polynomial, there are points where the derivative of sine equals zero.

Key Concepts

Polynomial Function

A polynomial function is a mathematical expression consisting of variables, coefficients, and exponents. It is constructed using a finite combination of algebraic terms. The degree of a polynomial is determined by the highest exponent of its variable. In our exercise, we are asked to construct a polynomial function that has specific zeros at the points:

• \(x=-2\)
• \(x=-1\)
• \(x=0\)
• \(x=1\)
• \(x=2\)

To achieve this, we can use the fact that a polynomial function can be expressed in its factored form. Each zero provided will give us a factor of the form \((x - a)\), where \(a\) is a zero. Thus, for our exercise:

• The polynomial is \(f(x) = (x+2)(x+1)x(x-1)(x-2)\).

This factored form is essential because it easily displays the zeros of the function. It demonstrates where the polynomial intersects the x-axis. This form is useful when analyzing the behavior of derivatives, as needed in the context of Rolle's Theorem.

Derivative

A derivative represents the rate at which a function is changing at any given point. For a polynomial function, the derivative is key to understanding how the function's graph behaves. We can find the derivative of a polynomial function by applying standard differentiation rules to each term. In our case, the polynomial \(f(x) = (x+2)(x+1)x(x-1)(x-2)\) needs to be differentiated to find its derivative \(f'(x)\).

• Differentiation simplifies each power of \(x\), reducing its exponent by one and multiplying its coefficient by the original exponent.
• The derivative provides us with the slope of the tangent line at each point of the function \(f(x)\).

Rolle's Theorem comes into play here, as it requires the derivative to be zero at least once between any two consecutive zeros of the function. This means that at those points, the tangent line to the curve would be horizontal, indicating possible local maxima or minima.

Zeros of a Function

The zeros of a function are the x-values at which a function evaluates to zero. For polynomials, these points are crucial because they divide the x-axis into different segments, each potentially possessing different characteristics in the curve's shape. For the polynomial \(f(x)\):

• The zeros are crucial as they determine the factors \((x+2), (x+1), x, (x-1), (x-2)\).
• Each zero represents a point where the graph of \(f(x)\) crosses the x-axis.

Rolle's Theorem depends on having consistent function values at both endpoints within a closed interval, making a role of zeros pivotal for identifying segments of interest. This is because between any two consecutive zeros of \(f(x)\), according to Rolle’s Theorem, there is at least one point where the derivative \(f'(x)\) becomes zero, ensuring the slope of the tangent to the graph is perfectly horizontal.

Continuous and Differentiable Functions

To apply Rolle's Theorem, the function in question must satisfy two important conditions: it must be continuous and differentiable over the interval considered. Continuity means there are no breaks, jumps, or holes in the function's graph over the interval. Differentiability means it is possible to compute a derivative at every point within the interval, indicating a smooth curve without sharp corners.

• The polynomial \(f(x) = (x+2)(x+1)x(x-1)(x-2)\) is both continuous and differentiable over all real numbers, including between its zeros.
• The sine function \(g(x) = \sin x\) is another great example. It is continuous and differentiable over all real numbers, including between any two consecutive zeros such as \(x=0\) and \(x=\pi\).

These properties ensure that Rolle’s Theorem can be applied, highlighting the existence of at least one point where the tangent slope equals zero between any two zeros of the function. Furthermore, they allow us to visualize and predict the behavior of functions as they traverse across their defined domains.