Hyperbolic cosine. The hyperbolic cosine function is defined by the equation $$ \cosh x=\frac{e^{x}+e^{-x}}{2} $$ Write a program to calculate the hyperbolic cosine of a user-supplied value \(x\). Use the program to calculate the hyperbolic cosine of \(3.0\). Compare the answer that your program produces to the answer produced by the MATLAB intrinsic function cosh \((x)\). Also, use MATLAB to plot the function \(\cosh (x)\). What is the smallest value that this function can have? At what value of \(x\) does it occur?

MATLAB is a powerful computing environment that not only allows for matrix manipulations, data analysis, and visualization but also integrates a high-level programming language.
When learning MATLAB programming, one of the key concepts to master is writing custom functions for specific mathematical operations, as exemplified by the hyperbolic cosine function, \( \cosh x \).

Writing Custom Functions in MATLAB

Just like the \texttt{hyperbolic\_cosine} function created in the exercise, MATLAB functions are defined using the \texttt{function} keyword followed by the output variable, the name of the function, and any input parameters. The body of the function encapsulates the computational logic. Here, the exponential function \texttt{exp} is utilized to calculate \( \cosh x \) based on its mathematical definition. Remember, personalized functions must be both efficient and accurately reflect the mathematical principles they embody to be useful in any computational task.
Following best practices, such as commenting code and performing validation checks, can greatly enhance the readability and reliability of MATLAB programming. These practices are crucial when creating robust scripts for engineering and scientific computations.

The inclusion of programming in the engineering curriculum equips students with the essential skills needed to analyze and solve complex engineering problems. MATLAB stands out as a preferred tool in this educational sphere due to its simplicity and the wide range of applications it supports.

Application of Programming in Problem Solving

Tasks such as calculating the hyperbolic cosine or any other mathematical functions, simulations, and data analysis are common in engineering coursework. For example, understanding how to write a function to compute \( \cosh x \) lays the foundation for tackling more challenging problems that involve numerical methods, optimization, or signal processing.
Furthermore, programming teaches students critical thinking and logical reasoning, which are indispensable skills in any engineering discipline. By comparing a custom MATLAB function to the built-in equivalent, as done with the hyperbolic cosine, students learn to validate their own solutions and gain trust in their programming prowess.

MATLAB excels in dealing with a wide array of mathematical functions that are central to engineering and scientific analysis. The hyperbolic cosine is just one example of the numerous intrinsic functions available in the software.

Understanding Hyperbolic Functions in MATLAB

Functions like \texttt{cosh}, \texttt{sinh}, and \texttt{tanh} are used to compute hyperbolic sine, cosine, and tangent, respectively. These functions can be explored through both custom implementations and MATLAB's built-in functions. This dual approach helps in understanding the theory behind the functions while also learning how to utilize the efficient built-in algorithms that MATLAB offers.
Plotting capabilities in MATLAB allow students to visualize these functions easily, as demonstrated in the hyperbolic cosine plot exercise. This not only reinforces the understanding of the function's behavior but also demonstrates that for \( \cosh x \), the smallest value it can have is 1, which occurs at \( x = 0 \). Visual aids like plots are incredibly influential in teaching complex mathematical concepts as they provide intuitive insights beyond numerical data.