2.1 - Mathematical functions
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Introduction to Built-in Functions
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Today, we are here to explore the built-in mathematical functions available in MATLAB. These functions simplify technical computations significantly. Can anyone tell me what some basic mathematical operations are?
Are functions like sine and cosine included?
What about exponential and logarithmic functions?
Absolutely! Functions such as `sin(x)`, `cos(x)`, and `exp(x)` are some examples of MATLAB's powerful offerings. Remember, we can categorize these as 'elementary functions' and they are very useful in engineering and physics.
How do we find these functions in MATLAB?
Good question! You can type `help elfun` for elementary functions and `help specfun` for special functions. Let's remember this by the acronym 'EFS' for Easy Function Search!
That’s helpful! What about the mathematical constants?
Great point! Constants like π and the imaginary unit are also built-in. For example, `pi` gives you the value of π directly in MATLAB.
To summarize, today we learned about key built-in functions, how to access them, and discussed some important constants like π. Next, let’s look at examples.
Practical Examples of Functions
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Now that we've reviewed the basics, let's explore some examples. For instance, calculating `y = exp(-a)*sin(x) + 10*sqrt(y)`. Any idea how we can do this?
We would first set values for `a`, `x`, and `y`, right?
Correct! Let’s code that in MATLAB. For example, setting `a = 5`, `x = 2`, and `y = 8` and executing it gives us an output of approximately `28.2904`. Remember that this reinforces the concept that we can handle various numerical data types like scalars, vectors, and even matrices in MATLAB.
What happens when we want to compute the logarithm?
Good question! MATLAB has two types: the natural logarithm `log(x)` and the base-10 logarithm `log10(x)`. For example, if you type `log(142)`, you get about `4.9558`; for `log10(142)`, it's `2.1523`.
So they are not the same!
Exactly! Understanding this will help you choose the right function based on your calculations. To wrap up, we practiced some essential computations and learned about varying logarithmic functions.
Common Pitfalls with Functions
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Before we conclude our lesson, let’s talk about some common pitfalls when using MATLAB functions. Can anyone guess what might be a mistake?
Maybe it’s reassigning built-in function names?
Yes! Reassigning a function could lead to unforeseen errors. Stick with the original names for built-in functions like `i` and `j` in MATLAB as they are preset for imaginary values. It’s better to use `ii` or `jj` as loop indices.
What if I accidentally overwrite them? What happens?
That's a critical error! To avoid confusion in your code, always keep built-in functions intact. Just remember the phrase 'Keep Built-ins Clean'!
Thanks for the tips! It sounds like following this will save me from headaches down the road.
Definitely! In summary, we highlighted essential practices when using MATLAB functions to help you avoid common mistakes. Now, let's move on to some quick exercises.
Introduction & Overview
Read summaries of the section's main ideas at different levels of detail.
Quick Overview
Standard
MATLAB offers a rich library of predefined mathematical functions essential for technical computing, including elementary functions and constants. The section emphasizes how functions like trigonometric, logarithmic, and exponential operations can be performed and provides examples for clarity.
Detailed
Overview of Mathematical Functions in MATLAB
MATLAB serves as a powerful tool for technical computing, equipped with a wide array of predefined mathematical functions. This section discusses the significance of built-in functions, such as trigonometric functions like sin(x), cos(x), and exponential functions like exp(x). Furthermore, it presents handy tables listing elementary functions and constants, such as π and the imaginary unit. Readers are guided through examples that illustrate how to compute complex mathematical expressions, enhancing their understanding of MATLAB's capabilities in handling both simple and advanced calculations.
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Introduction to Mathematical Functions in MATLAB
Chapter 1 of 6
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Chapter Content
MATLAB offers many predefined mathematical functions for technical computing which contains a large set of mathematical functions.
Typing help elfun and help specfun calls up full lists of elementary and special functions respectively.
Detailed Explanation
MATLAB is a powerful programming environment that provides users with a variety of mathematical functions that can be used for computations. These functions are predefined, meaning they are built into MATLAB and ready to use without any additional programming. To see a full list of elementary functions, you can use the commands 'help elfun' and for special functions, 'help specfun'. This helps users understand the capabilities of MATLAB in handling mathematical operations.
Examples & Analogies
Think of MATLAB like a toolbox for a mechanic. Just as a mechanic has a variety of tools available to fix cars, MATLAB has a toolbox filled with functions that can help complete various mathematical tasks. By looking up the tools (or functions) it has, users can solve their mathematical problems efficiently.
Built-in Mathematical Functions
Chapter 2 of 6
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There is a long list of mathematical functions that are built into MATLAB. These functions are called built-ins. Many standard mathematical functions, such as sin(x), cos(x), tan(x), ex, ln(x), are evaluated by the functions sin, cos, tan, exp, and log respectively in MATLAB. Table 2.1 lists some commonly used functions, where variables x and y can be numbers, vectors, or matrices.
Detailed Explanation
MATLAB includes a variety of built-in mathematical functions that users can utilize. These functions cover a range of basic mathematical operations, such as trigonometric functions (like sine, cosine, and tangent), exponential functions, and logarithmic functions. The special variable types in MATLAB mean that users can work with not only single numbers but also arrays (vectors and matrices) to perform calculations on multiple values at the same time.
Examples & Analogies
Imagine you are baking cookies, and you need to measure out different ingredients. The built-in functions in MATLAB are like the measuring cups and spoons that allow you to accurately measure sugar, flour, and butter (the numbers) all at once, efficiently mixing them to create perfect cookie dough (the results of your calculations).
Elementary Functions Table
Chapter 3 of 6
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Chapter Content
Table 2.1: Elementary functions
cos(x) Cosine
abs(x) Absolute value
sin(x) Sine
sign(x) Signum function
tan(x) Tangent
max(x) Maximum value
acos(x) Arc cosine
min(x) Minimum value
asin(x) Arc sine
ceil(x) Round towards +∞
atan(x) Arc tangent
floor(x) Round towards −∞
exp(x) Exponential
round(x) Round to nearest integer
sqrt(x) Square root
rem(x) Remainder after division
log(x) Natural logarithm
angle(x) Phase angle
log10(x) Common logarithm
conj(x) Complex conjugate
Detailed Explanation
The table presents a selection of elementary functions that are readily available in MATLAB. Each function serves a specific mathematical purpose. For instance, 'cos(x)' computes the cosine of an angle in radians, 'abs(x)' gives the absolute value of a number, and 'sqrt(x)' calculates the square root. Users need to familiarize themselves with these functions to efficiently implement them in their calculations in MATLAB.
Examples & Analogies
Think of these functions as different types of calculators for various tasks. Just as you might use a scientific calculator for advanced mathematics (like calculating sine or cosine) and a basic calculator for simple arithmetic (like finding maximum or minimum values), MATLAB provides a comprehensive set of tools that can handle both complex and simple mathematical operations with ease.
Predefined Constants
Chapter 4 of 6
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Chapter Content
In addition to the elementary functions, MATLAB includes a number of predefined constant values. A list of the most common values is given in Table 2.2.
Table 2.2: Predefined constant values
pi The π number, π = 3.14159...
i,j The imaginary unit i, √1−
Inf The infinity, ∞
NaN Not a number
Detailed Explanation
MATLAB also provides a collection of predefined constants that are frequently used in mathematical calculations. For example, pi is a constant representing the value of π, and it can be directly used in mathematical expressions. The imaginary unit i is also predefined in MATLAB, which allows users to seamlessly work with complex numbers. Additionally, constants such as Inf (infinity) and NaN (not a number) help in representing mathematical concepts that cannot be represented by regular numbers.
Examples & Analogies
Think of these constants like common ingredients in a recipe. Just as you might always need flour or sugar when baking, certain constants like π and the imaginary unit are frequently needed in mathematical formulas. By having these constants predefined, MATLAB saves time—just like having your ingredients pre-measured and ready to go allows you to bake faster.
Examples of Basic Mathematical Functions
Chapter 5 of 6
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Chapter Content
We illustrate here some typical examples which relate to the elementary functions previously defined.
As a first example, the value of the expression y = e^(asin(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
The subsequent examples are
log(142)
ans = 4.9558log10(142)
ans = 2.1523
Note the difference between the natural logarithm log(x) and the decimal logarithm (base 10) log10(x).
Detailed Explanation
In this section, we find concrete examples of how to use the functions discussed earlier. We calculate a more complex expression that involves the sine function, the exponential function, and the square root. This example shows how MATLAB can handle various calculations in one line of code. Additionally, we explore logarithmic functions, illustrating the difference between natural logarithms and common logarithms. This highlights the importance of understanding which function to use depending on the mathematical scenario.
Examples & Analogies
Imagine you are a scientist conducting an experiment. The equations you use are like the recipes for mixing your chemicals. With MATLAB, you can quickly calculate the outcomes based on the variables you have, just as a scientist would mix known quantities to observe a reaction. For example, the expression y = e^(asin(x)) + 10√y could represent a calculation for determining a compound's reaction based on different factors.
Important Notes on Functions
Chapter 6 of 6
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Chapter Content
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.
Detailed Explanation
This notes section provides important guidelines when using functions in MATLAB to prevent potential issues. It emphasizes the importance of using built-in functions as intended and avoiding unintended variable assignments to those functions. Moreover, it suggests that while MATLAB uses i and j for imaginary units, they can also be used as loop indices, which might confuse users. Thus, it's better to use alternative letter combinations if you're creating loops within your code.
Examples & Analogies
Imagine organizing a team to finish a project. If you assign team members names that are too similar or the same as project names, it can lead to confusion. By taking steps to clearly differentiate between individual responsibilities and project tasks, you prevent mix-ups. Similarly, following MATLAB's guidelines helps maintain clarity in your calculations and avoids conflicts within your code.
Key Concepts
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Built-in Functions: Essential for computations in MATLAB.
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Elementary Functions: Include trigonometric, logarithmic, and exponential functions.
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Constants: Fixed values in MATLAB such as π and the imaginary unit.
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Natural vs. Common Logarithm: Key difference in their bases.
Examples & Applications
Example 1: Calculating sin(π/4) in MATLAB results in approximately 0.7071.
Example 2: The exponential function exp(10) gives approximately 22026.
Memory Aids
Interactive tools to help you remember key concepts
Rhymes
For sin and cosine, use sin and cos, In MATLAB they perform without a pause.
Stories
Imagine you are a wizard, casting spells with numbers. With sin(x) and cos(x), make waves dance on the water!
Memory Tools
Remember 'ECL' for Elementary Functions: Exponential, Cosine, Logarithm.
Acronyms
Use 'BIE'
Built-in
Important
Efficient when recalling fundamental functions.
Flash Cards
Glossary
- Builtin Functions
Predefined functions in MATLAB that perform specific mathematical operations.
- Elementary Functions
Basic mathematical functions such as sine, cosine, and exponential that can be directly used in MATLAB.
- Constants
Standard fixed values such as π and i (the imaginary unit) that are readily available in MATLAB.
- Logarithm
The inverse operation to exponentiation, returning the exponent as a result.
Reference links
Supplementary resources to enhance your learning experience.