Problem 9 - Calculus Early Transcendentals

Question

(a) Find an interval \([a, b]\) on which $$ f(x)=x^{4}+x^{3}-x^{2}+x-2 $$ satisfies the hypotheses of Rolle's Theorem. (b) Generate the graph of \(f^{\prime}(x),\) and use it to make rough estimates of all values of \(c\) in the interval obtained in part (a) that satisfy the conclusion of Rolle's Theorem. (c) Use Newton's Method to improve on the rough estimates obtained in part (b).

Step-by-Step Solution

Answer

Interval found: \([-2, 1]\); estimated roots: around \(-1.5, -0.5, 0.5\); refine using Newton's Method.

Step 1: Finding the Interval

Rolle's Theorem states that if a function \( f \) is continuous on \([a, b]\), differentiable on \((a, b)\), and \( f(a) = f(b) \), then there exists at least one \( c \) in the interval \((a, b)\) such that \( f'(c) = 0 \). To find such an interval, we check where \( f(x) \) is continuous and differentiable. Since \( f(x) = x^4 + x^3 - x^2 + x - 2 \) is a polynomial, it is continuous and differentiable everywhere. We need to find an interval \([a, b]\) such that \( f(a) = f(b) \). Trying \( x = -2 \) and \( x = 1 \):\[ f(-2) = (-2)^4 + (-2)^3 - (-2)^2 + (-2) - 2 = 16 - 8 - 4 - 2 - 2 = 0 \]\[ f(1) = 1^4 + 1^3 - 1^2 + 1 - 2 = 1 + 1 - 1 + 1 - 2 = 0 \]Thus, \([a, b] = [-2, 1]\) is an appropriate interval.

Step 2: Graphing the Derivative

To apply Rolle's Theorem, we now need to consider the derivative \( f'(x) \) of the function. Calculate the derivative:\[ f'(x) = 4x^3 + 3x^2 - 2x + 1 \]Graph \( f'(x) \) to estimate where it crosses zero within \((-2, 1)\). Use graphing software or graphing calculator to plot \( f'(x) \) and check for zeros, which typically occur around \( x \approx -1.5, x \approx -0.5, \text{and} \ x \approx 0.5 \).

Step 3: Applying Newton's Method for Refinement

Newton's Method is an iterative numerical method used to refine roots of a real-valued function starting from an initial guess. It uses the formula:\[ x_{n+1} = x_n - \frac{f'(x_n)}{f''(x_n)} \]First, compute the second derivative:\[ f''(x) = 12x^2 + 6x - 2 \]Using one initial guess, say \( x_0 = -1.5 \), compute:\[ x_1 = x_0 - \frac{4(-1.5)^3 + 3(-1.5)^2 - 2(-1.5) + 1}{12(-1.5)^2 + 6(-1.5) - 2} \]Complete similar calculations for \( x_0 = -0.5 \) and \( x_0 = 0.5 \), using successive approximations until convergence.

Key Concepts

Polynomial Functions

Polynomials are algebraic expressions made up of terms consisting of a variable raised to a non-negative integer power, along with coefficients. In the context of this exercise, we are dealing with a polynomial function given by:

• \( f(x) = x^4 + x^3 - x^2 + x - 2 \)

Polynomial functions are particularly important in calculus due to their smooth and continuous nature. This means they are both continuous and differentiable everywhere on the real number line.
These properties make them ideal candidates for analysis using theorems like Rolle's Theorem. Rolle's Theorem requires the function to be continuous on a closed interval \([a, b]\) and differentiable on the open interval \((a, b)\). Since polynomial functions are naturally continuous and differentiable on their domain, they perfectly satisfy these conditions.
The problem in this exercise exploits these characteristics to identify an interval where the function's start and end values are equal, allowing us to apply Rolle's Theorem. By evaluating \( f(-2) \) and \( f(1) \), we confirm that the function outputs zero at both points, establishing the interval \([-2, 1]\) where Rolle's Theorem can be applied.

Newton's Method

Newton's Method is a practical numerical technique used to approximate the roots of a real-valued function. It is an iterative process, which means it involves repeating a certain calculation until we find a value that is close enough to the root. In simple terms, it works by taking an initial guess and refining it to get closer to the actual root.
To use Newton's Method, we apply the formula:

• \[ x_{n+1} = x_n - \frac{f'(x_n)}{f''(x_n)} \]

where \( x_n \) is our current estimate, \( f'(x_n) \) is the derivative of our function evaluated at \( x_n \), and \( f''(x_n) \) is the second derivative evaluated at the same point. This method requires us to know both the first and second derivatives of the function.
In the exercise, after estimating the roots obtained by graphing the derivative, such as \( x \approx -1.5, -0.5 \), and \( 0.5 \), we refine these estimates. Starting with our initial guesses, each iteration brings our \( x_n \) closer to the actual zero of the derivative, yielding more precise solutions. Newton's Method's power comes from its ability to converge quickly to a solution when the starting point is chosen wisely.

Graph of Derivative

Understanding the graph of a function's derivative gives us insights into the function's behavior, particularly where it increases or decreases. Determining where a polynomial's derivative is zero is central to finding critical points—and in the context of Rolle's Theorem, it helps identify points \( c \) where \( f'(c) = 0 \).
The derivative of our function, \( f(x) = x^4 + x^3 - x^2 + x - 2 \), is given by:

• \( f'(x) = 4x^3 + 3x^2 - 2x + 1 \)

Graphing this derivative involves plotting \( f'(x) \) over a chosen interval, such as \([-2, 1]\), to visually estimate where it crosses the x-axis, indicating potential critical points. As seen from the graphing step in the solution, the graph suggests zeros near \( x \approx -1.5, -0.5, \) and \(0.5\).
This graphical insight acts as a preliminary step, providing estimates before applying more precise methods like Newton's Method to verify and refine these critical points. By analyzing the graph, students gain a deeper understanding of the interplay between a function and its derivatives, enhancing both conceptual understanding and computational skill.