91Ó°ÊÓ

Calculating a derivative is a crucial aspect of understanding how functions change at any given point. In this exercise, we begin by dealing with the function \(f(x) = x^2 - \frac{2}{x}\). To find its derivative, \(f'(x)\), we perform differentiation, which gives us a new function expressing how \(f(x)\) changes as \(x\) changes.
In our function, differentiating \(x^2\) results in \(2x\), whereas the derivative of \(-\frac{2}{x}\) is \(\frac{2}{x^2}\) because of the power rule. Combining these, we're left with:
\[f'(x) = 2x + \frac{2}{x^2}\]
This derivative tells us the rate at which our original function's value is changing at any specific \(x\) value. Derivatives are a fundamental tool in calculus. They help us understand the behavior of graphs and optimize functions in various fields like physics and engineering.

The slope of a line represents how steep the line is, indicating the rate of change between two points. In this scenario, we seek the slope \(m\) of the line connecting the points \((1, -1)\) and \((2, 3)\).
To compute \(m\), we use the formula:
\[m = \frac{f(b) - f(a)}{b - a} = \frac{3 - (-1)}{2 - 1} = 4\]
Here's what happens:

• Subtract the y-values, which provides the change in the y-direction.
• Subtract the x-values for the x-direction change.
• Divide these results to find the slope \(m = 4\).

Understanding slope is key when interpreting graphs, as it determines how the function behaves. Knowing the slope helps predict future values and understand past behavior.

Convergence criteria are essential when employing iterative methods, like the Newton-Raphson method, to ensure we reach a stable solution. In our case, we are finding a \(c\)-value such that \(f'(c) = 4\).
Using the Newton-Raphson formula:
\[c_{n+1} = c_n - \frac{f'(c_n) - 4}{f''(c_n)}\]
The idea is to start with an initial guess \(c_0\), and update it iteratively until the sequence \(c_n\) converges. Convergence implies that successive values of \(c_n\) become practically identical, indicating stability and accuracy. To check for convergence, observe if \(c_{n+1} - c_n\) is less than a chosen small value (like 0.001).
This small threshold ensures high precision in our results. Iterative methods without convergence criteria might yield incorrect conclusions due to non-stability.

The second derivative, \(f''(x)\), offers more insight into the geometry of the function graph, such as concavity and inflection points. For our function \(f(x) = x^2 - \frac{2}{x}\), we've already calculated \(f'(x) = 2x + \frac{2}{x^2}\).
To find \(f''(x)\), we differentiate \(f'(x)\):
\[f''(x) = \frac{d}{dx}(2x) + \frac{d}{dx}(\frac{2}{x^2}) = 2 - \frac{4}{x^3}\]
Here, the second derivative tells us about the acceleration of the slope. It's particularly useful in the Newton-Raphson method to adjust the slope for iteration, helping us refine our approximation of \(c\). Understanding \(f''(x)\) is vital in various applications, from determining forces in physics to predicting price changes in economics.