Welcome to the first lesson of our course, "Optimization with SciPy." In this lesson, we're going to lay the foundation by learning how to define functions in Python. Functions are crucial in programming as they help you organize and reuse code efficiently. As we progress in the course, you'll see how understanding functions is vital for leveraging the full power of SciPy.

Let's quickly recall python functions, as we are going to use them a lot in this course.

In Python, you define a function using the def keyword, followed by the function name and parentheses (). Inside the parentheses, you can define parameters, which are inputs to the function. Let's start with a simple function that adds two numbers:

• def add_numbers(a, b): defines a function named add_numbers that takes two parameters, a and b.
• return a + b calculates the sum of a and b and returns the result.
• When you call this function with add_numbers(3, 4), it will return 7.

Let's now explore more complex functions.

Here's how to define $f(x) = x^2 + 4x + 4$:

• This function calculates the value of a quadratic expression x^2 + 4*x + 4.
• The ** operator is used to denote x raised to the power of 2.
• When you call this function with quadratic_function(2), it will return 16.

Let's define $f(x) = \sin{x} + \cos{x}$

• np.sin(x) and np.cos(x) use NumPy library functions to calculate the sine and cosine of x.
• The function returns the sum of the sine and cosine of x.
• When you call this function with trigonometric_function(np.pi/4), it will return approximately 1.4142.

Let's see another example – a definition of $f(x) = e^x - 3x$

• np.exp(x) computes the exponential of x, e^x, using the NumPy library.
• The function then subtracts 3*x from the exponential and returns the result.
• When you call this function with exponential_function(1), it will return approximately -0.2817.

Here is your last example. Let's define $f(x) = \ln{x} + x$

• np.log(x) computes the natural logarithm of x.
• The function checks if x is positive, as the logarithm is undefined for non-positive values, and if not, it returns None.
• When you call this function with logarithmic_function(2), it will return approximately 1.6931.

In this lesson, you learned how to define, test, and use functions in Python. You explored different types of mathematical functions and saw how to implement them using Python's syntax. Understanding these principles is crucial as functions will be a cornerstone for optimizing mathematical problems with SciPy.

Now, as you progress to the practice exercises, remember to apply these concepts by experimenting with various function definitions. Test them with different inputs to solidify your understanding. Happy coding, and enjoy the journey of exploring optimization with SciPy!