Problem 3 - Single Variable Calculus

Question

Use Newton's method to find the zero(s) of \(f\) to four decimal places by solving the equation \(f(x)=0 .\) Use the initial estimate \((s) x_{0}\). \(f(x)=\frac{3}{2} x^{4}-2 x^{3}-6 x^{2}+8, \quad x_{0}=1\) and \(x_{0}=3\)

Step-by-Step Solution

Answer

Using Newton's method with the given function, $f(x) = \frac{3}{2}x^4 - 2x^3 - 6x^2 + 8$, and its derivative, \(f'(x) = 6x^3 - 6x^2 - 12x\), we iterate starting from both initial estimates, \(x_0=1\) and \(x_0=3\). After iterating until convergence to four decimal places, we find two distinct zeros of the function: For \(x_0=1\), the zero is approximately \(x \approx 0.4926\). For \(x_0=3\), the zero is approximately \(x \approx 2.1646\).

Step 1: Find the derivative of the function.

Differentiate the given function, \(f(x)=\dfrac{3}{2}x^4 - 2x^3 - 6x^2 + 8\), with respect to x: \[ f'(x) = 6x^3 - 6x^2 - 12x \]

Step 2: Newton's Method Iteration

Apply Newton's method iterative formula, using the initial estimates \(x_0=1\) and \(x_0=3\). Repeat the process until the estimated zero converges to four decimal places. The iteration formula for Newton's method is: \[ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \]

Step 3: Iterate Starting with \(x_0=1\)

Estimate the zero with the initial estimate of \(x_0=1\). Continue iterating using the Newton's method formula until the value of \(x\) converges to four decimal places.

Step 4: Iterate Starting with \(x_0=3\)

Estimate the zero with the initial estimate of \(x_0=3\). Continue iterating using the Newton's method formula until the value of \(x\) converges to four decimal places.

Step 5: Interpret Results

The values of \(x_n\) obtained from each initial estimate \(x_0\) will converge to the zeros of the given function, \(f(x)\), in each case when they are not changing more than \(0.0001\). Once you have completed the iterations for both initial estimates, compare and/or round the results to four decimal places to find the distinct zero(s) of \(f(x)\).

Key Concepts

Zero of a Function

Finding the zero of a function means identifying a point where the function crosses the x-axis. This is where the output of the function is zero. In mathematical terms, if we have a function \(f(x)\), a zero of this function is a value \(x = a\) for which \(f(a) = 0\).

Finding these points is crucial as they represent the solutions to the equation \(f(x) = 0\). Identifying zeros is especially important in algebra and calculus, as they can tell us about the behavior and characteristics of the function.

In the exercise, we aim to find these zeros using Newton's method, which provides an efficient way to solve the equation when a good initial guess is available.

Derivative

The derivative of a function provides information about the rate at which the function is changing at any given point. It is represented as \(f'(x)\) for a function \(f(x)\).

The concept of a derivative is fundamental in calculus and reformulates the idea of a slope for curves. In the context of Newton's method, the derivative is crucial because it helps us refine our guesses towards the actual zero of the function.

Specifically, the derivative \(f'(x)\), tells us how steep the tangent to the function is at any point. This insight into the slope is used in Newton's iteration formula to make successive approximations. For this exercise, the derivative was calculated as \(f'(x) = 6x^3 - 6x^2 - 12x\). This expression will be employed to adjust our guesses through iterations.

Iteration

Iteration is the process of repeating a set of operations until a specific condition is met. In Newton's method, iteration is used to successively refine guesses for the zeros of a function.

Newton's iterative formula is given by \(x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}\). This formula uses the current guess \(x_n\) and refines it using the formula, thus providing a new guess \(x_{n+1}\). This process continues until we reach our desired accuracy.

• The iteration starts with an initial estimate (\(x_0\)), which is an educated guess of where the zero might be.
• The next estimate \(x_1\) is calculated using the iterative formula.
• This process is repeated, calculating \(x_2\), \(x_3\), and so on.

By continuing the iterations, the value of \(x\) becomes closer to the actual zero.

Convergence

Convergence in the context of Newton's method refers to the process where the successive estimates for the zero of the function tend to become stable or change only slightly with each iteration.

When applying Newton's method, convergence is achieved when the difference between two successive estimates is smaller than a predetermined tolerance level, such as 0.0001 as used in this exercise. This small difference indicates that further iterations will make negligible changes, suggesting that a zero has been identified with the desired precision.

• Convergence is critical because it indicates that the method is successful and the result is reliable.
• It provides a stopping criterion for the iterations, ensuring that we do not continue indefinitely.

After achieving convergence, the results can be rounded to the required decimal places, confirming the calculations have reached sufficient accuracy.