Suppose we have a function \(f(x)\) and we want to calculate its derivative at a point \(x\). We can do that with pencil and paper if we know the mathematical form of the function, or we can do it on the computer by making use of the definition of the derivative: $$ \frac{\mathrm{d} f}{\mathrm{~d} x}=\lim _{\delta \rightarrow 0} \frac{f(x+\delta)-f(x)}{\delta} $$ On the computer we can't actually take the limit as \(\delta\) goes to zero, but we can get a reasonable approximation just by making \(\delta\) small. a) Write a program that defines a function \(f(x)\) returning the value \(x(x-1)\), then calculates the derivative of the function at the point \(x=1\) using the formula above with \(\delta=10^{-2}\). Calculate the true value of the same derivative analytically and compare with the answer your program gives. The two will not agree perfectly. Why not? b) Repeat the calculation for \(\delta=10^{-4}, 10^{-6}, 10^{-8}, 10^{-10}, 10^{-12}\), and \(10^{-14}\). You should see that the accuracy of the calculation initially gets better as \(\delta\) gets smaller, but then gets worse again. Why is this? We will look at numerical derivatives in more detail in Section 5.10, where we will study techniques for dealing with these issues and maximizing the accuracy of our calculations.

Step by step solution

01

Define the Function

First, define the function in Python. The function given is \( f(x) = x(x-1) \). In Python, it can be written as:```pythondef f(x): return x * (x - 1)```

02

Write a Function for Numerical Derivative

Write a function that takes \( f(x) \), a point \( x \) and \( \delta \), and computes the derivative using the formula. This can be written as:```pythonimport numpy as npdef numerical_derivative(f, x, delta): return (f(x + delta) - f(x)) / delta```

03

Calculate the Derivative at x using δ = 10^-2

Calculate the derivative of \( f(x) \) at \( x = 1 \) using \( \delta = 10^{-2} \). Here's the code to do this:```pythonx = 1delta = 10**-2numerical_derivative_value = numerical_derivative(f, x, delta)print('Numerical Derivative:', numerical_derivative_value)```

04

Calculate the True Derivative Analytically

Find the true derivative of \( f(x) \) analytically. The derivative of \( f(x) = x(x-1) \) is obtained using the power rule:\[ f'(x) = 2x - 1 \]Substitute \( x = 1 \):\[ f'(1) = 2(1) - 1 = 1 \]

05

Compare Numerical and Analytical Results

Compare the output from the program with the analytical result. The code could output something like:```pythontrue_derivative_value = 2 * x - 1print('True Derivative:', true_derivative_value)```The numerical and analytical results should be close but not identical.

06

Explain Initial Discrepancy

The discrepancy is because the numerical method is an approximation that depends on the chosen \( \delta \). As \( \delta \rightarrow 0 \), the approximation should get closer to the true value.

07

Calculate for Different δ Values

Repeat the calculation for various \( \delta \) values: \( 10^{-4}, 10^{-6}, 10^{-8}, 10^{-10}, 10^{-12}, 10^{-14} \). Populating the results can be done with a loop:```pythonfor exp in range(2, 15, 2): delta = 10**-exp numerical_derivative_value = numerical_derivative(f, x, delta) print(f'Delta: {delta}, Numerical Derivative: {numerical_derivative_value}')```Observe how the accuracy changes.

08

Explain the Behavior with Different δ Values

As \( \delta \) gets smaller, the accuracy improves initially. However, for very small \( \delta \), numerical errors due to the limitations of floating-point precision in computers cause the accuracy to degrade. This results in incorrect derivatives for extremely small values of \( \delta \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Computational Physics

Computational physics is a field that combines physics and computer science to solve complex physical problems. Unlike theoretical physics, which relies on pen-and-paper calculations, computational physics uses algorithms and numerical simulations to understand physical systems. For example, calculating the derivative of a function numerically, as in the given exercise, is a common task. Numerical methods, like the one used to approximate derivatives, play a significant role in this field. Computational physicists often deal with large datasets and complex simulations that would be impossible to solve analytically. Computers help them to visualize these systems and understand underlying physical laws.

Python Programming

Python is a versatile and beginner-friendly programming language, widely used in computational physics for its simplicity and powerful libraries. In the exercise, we defined a function and computed its derivative using Python.

First, we defined the function as:
```python
def f(x):
return x * (x - 1)
```
Next, we wrote a function to compute the numerical derivative:
```python
import numpy as np
def numerical_derivative(f, x, delta):
return (f(x + delta) - f(x)) / delta
```
Python's simplicity allows you to focus on the problem at hand rather than the intricacies of the programming language.

Calculating the derivative at a specific point, like we did in the exercise, can be achieved with a few lines of code. Its extensive libraries, such as numpy, provide additional tools for numerical calculations, making Python an excellent choice for computational tasks.

Floating-point Precision

Floating-point precision is a crucial concept in numerical computations. Computers represent real numbers using a fixed number of bits, which means they can't represent numbers exactly but rather as approximations. This limitation is known as floating-point precision.

In our exercise, we observed that as \( \delta \) gets smaller, our calculation accuracy improves only up to a point. When \( \delta \) becomes very small, the difference between the function values, \( f(x + \delta) \) and \( f(x) \), can be too tiny relative to the numerical precision of the computer. This causes significant round-off errors.

These errors stem from the limitations of how computers handle floating-point arithmetic, such as how numbers are rounded and the inherent inaccuracies in the representation of very small or very large numbers. When performing numerical differentiation, choosing an appropriate \( \delta \) is crucial to balance between accuracy and floating-point error.

Numerical Methods

Numerical methods are algorithms used to solve mathematical problems that cannot be solved analytically. In our exercise, we used a simple numerical method to approximate the derivative of a function.

A common numerical method for derivatives is the finite difference method, which approximates the derivative by calculating the difference between function values at two nearby points. The formula we used, \[ \frac{f(x+\delta)-f(x)}{\delta} \], is a basic example of this method.

Though straightforward, numerical methods often require careful consideration of factors like the step size (\delta) and error analysis. More sophisticated methods exist to increase accuracy and minimize numerical errors, such as central difference or higher-order techniques. These methods are essential for solving complex problems in science and engineering where analytical solutions are impossible.