2.1.1. Examples

As a first example, the value of the expression y = e^(-a)sin(x) + 10√y, for a = 5, x = 2, and y = -8 is computed by

a = 5; x = 2; y = 8; y = exp(-a)sin(x) + 10sqrt(y) y = 28.2904

In this example, we're calculating a complex mathematical expression involving the exponential function, sine function, and square root. First, we assign values to 'a', 'x', and 'y', where 'a' is 5, 'x' is 2, and 'y' is 8. The command 'exp(-a)' computes the value of e raised to the power of -5. The 'sin(x)' function gives us the sine of 2 radians. The square root of 'y' is obtained using 'sqrt(y)', which in this case is the square root of 8. Finally, we combine these computations following the given formula to arrive at the result of approximately 28.2904.

Think of this example like calculating a recipe. You need specific amounts of your ingredients (values for 'a', 'x', and 'y'), and you follow a recipe (the mathematical formula) to create a dish (the final result). Just like measuring out spices or baking soda, each mathematical function in MATLAB helps you mix together the 'ingredients' to get the right 'taste' or answer.

The subsequent examples are

log(142) ans = 4.9558 log10(142) ans = 2.1523 Note the difference between the natural logarithm log(x) and the decimal logarithm (base 10) log10(x).

Here, we demonstrate calculations involving logarithms. When you input 'log(142)', MATLAB gives you the natural logarithm of 142, which is approximately 4.9558. This represents the power to which e (approximately 2.718) must be raised to obtain 142. When you use 'log10(142)', you receive the logarithm of 142 to base 10, which is about 2.1523. This means 10 must be raised to approximately 2.1523 to produce 142. Understanding the difference between these two logarithms is essential because they are used in various fields of mathematics and science.

Consider logarithms like measuring how many folds of paper you need to get a certain height; each logarithmic base is like a different type of folding technique. The natural logarithm folds in a more 'natural' way (using base e), while the common logarithm is like using a conventional method (base 10). Each method gets you to the height (or value) but might require a different number of folds.

To calculate sin(π/4) and e^10, we enter the following commands in MATLAB,

sin(pi/4) ans = 0.7071 exp(10) ans = 2.2026e+004

In this part, we calculate the sine of π/4 and the exponential function for the number 10. When we enter 'sin(pi/4)', MATLAB computes the sine of 45 degrees (or π/4 radians), which is 0.7071. Similarly, 'exp(10)' calculates e raised to the power of 10, yielding approximately 22026. This result shows how rapidly e^x grows as 'x' increases. Such calculations are commonly used in fields such as physics and engineering due to their applications in various formulas.

Imagine you're at a carnival. The sine function calculates how high you’d go if you were climbing a Ferris wheel, while the exponential function measures how quickly you could run up that Ferris wheel if it were spiraling up really fast! Just like there are different rides, each mathematical function gives you different results based on the 'ride' you're interested in.

Notes: Only use built-in functions on the right-hand side of an expression. Reassigning the value to a built-in function can create problems.

There are some exceptions. For example, i and j are pre-assigned to √(-1). However, one or both of i or j are often used as loop indices.

To avoid any possible confusion, it is suggested to use instead ii or jj as loop indices.

This section offers several important reminders about using built-in functions in MATLAB. Built-in functions, like sin, exp, and log, should not be reassigned to any variable because this can lead to errors. This is crucial as it maintains the integrity of the functions. The exception to this rule is the use of 'i' and 'j', which are usually used to denote the imaginary unit but can also serve as variables in loops. To minimize confusion, it's recommended to use different variable names, such as 'ii' or 'jj', when working with loops.

Think of built-in functions as kitchen appliances. If you label your toaster as a 'coffee maker', you'd get confused when trying to make toast! Similarly, keep built-in functions intact so they work correctly. Using 'i' and 'j' as variables is like borrowing your friend's well-known coffee cup for your drinks: it's best to use your own unique cups (like 'ii' or 'jj') to avoid mix-ups.

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