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MATLAB Mathematical Functions
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Today, we're diving into MATLAB's mathematical functions. Who can remind me what types of mathematical functions we might need in technical computing?
We need functions for sin, cos, logarithms, and maybe some constants?
Exactly! MATLAB provides built-in functions for elementary operations like `sin(x)`, `exp(x)`, and many more. To explore these, you can type `help elfun` for elementary functions and `help specfun` for special functions. Let's remember the acronym **ELA** to recall those functions: Elementary, Logarithmic, and Algebraic!
What do you mean by special functions?
Good question! Special functions include more complex operations that aren't usually covered by basic functions. They're essential for advanced mathematical computations.
Can we see examples of these functions in action?
Certainly! For example, consider calculating `y = exp(-a)*sin(x) + 10*sqrt(y)`. If we assign values, we can compute this in MATLAB.
Is it correct to reassign the output to a built-in function?
No, avoid that! It can lead to confusion in your code. Let's summarize: Always use built-ins on the right side!
Basic Plotting in MATLAB
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Let's pivot to the plotting capabilities of MATLAB. Why do you think visualizing data is important?
Visuals help us understand and interpret data better!
That's right! To create a basic plot, you need vectors for x and y points. If I want to plot the sine function, what do you think my x vector would look like?
It should start at 0 and go to 2Ο.
Excellent! In MATLAB, you can create that vector with `x = 0:pi/100:2*pi`. Then, compute `y = sin(x)` and visualize with `plot(x,y)`. Can anyone spot what happens if we omit the step size?
It defaults to an increment of 1!
Spot on! After plotting, how do we add titles and labels?
Use `xlabel`, `ylabel`, and `title` commands to annotate the plot.
Exactly! Annotations make graphs comprehensive. Remember to apply your `style_color_marker` for unique visuals in your plots.
Multiple Data Sets
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Now, can anyone tell me how we might plot multiple datasets together in MATLAB?
We can use `plot` with multiple x and y pairs!
Correct! Letβs say we want to plot `y1 = 2*cos(x)`, `y2 = cos(x)`, and `y3 = 0.5*cos(x)`. We use a command like `plot(x,y1,'--',x,y2,'-',x,y3,':')`. What do the symbols mean?
They represent different line styles!
Exactly! Customizing visual styles helps distinguish datasets. Furthermore, which command would we use to include a legend in our plot?
`legend` command!
Great job! By summarizing our plots with legends and titles, we enhance clarity. Let's not forget to review axis limits with the `axis` command too.
Introduction & Overview
Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.
Quick Overview
Standard
In this section, readers will learn about MATLAB's built-in mathematical functions, including examples and predefined constants. It also covers basic plotting techniques for visualizing data using MATLAB, emphasizing the importance of graphics in understanding mathematical equations.
Detailed
Detailed Summary
This section lays the foundation for using MATLAB in technical computing. It begins with an overview of mathematical functions, outlining how to access lists of predefined functions using commands like help elfun and help specfun. The text describes key mathematical functions such as trigonometric functions, logarithms, and their corresponding MATLAB commands, enhancing the readerβs ability to perform calculations with ease.
Furthermore, it discusses the importance of these functions in practical examples, demonstrating how to compute values effectively. The section is enriched with tables listing elementary functions and predefined constant values such as Ο and the imaginary unit.
After establishing the mathematical groundwork, the section transitions to basic plotting techniques. Readers will learn how to create simple 2D plots by first defining vectors for x and y coordinates and then utilizing the plot function to visualize data. It describes how to annotate plots with titles and axis labels, and it introduces methods for displaying multiple datasets within a single graph. Additional details on customizing line styles and colors enhance the visual aspect of data presentation.
The chapter closes with a focus on MATLAB's ability to combine technical computing and graphical representation, allowing for clearer understanding and communication of mathematical concepts.
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Mathematical Functions in MATLAB
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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.
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), e^x, ln(x), are evaluated by the functions sin, cos, tan, exp, and log respectively in MATLAB.
Detailed Explanation
In MATLAB, there are many built-in mathematical functions that you can use for calculations. These functions save you time by eliminating the need to write the mathematical formulas from scratch. For example, if you need to calculate the sine of an angle, instead of calculating it manually using a formula, you can use the built-in function sin(). You can list all available functions in MATLAB for standard operations by typing help elfun for elementary functions and help specfun for special functions in the MATLAB command window.
Examples & Analogies
Think of the built-in functions like tools in a toolbox. Just as you would grab a hammer instead of trying to create one yourself every time you need to drive a nail, you use these predefined functions instead of trying to compute sine or logarithm using basic principles each time.
Common Mathematical Functions
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Table 2.1 lists some commonly used functions, where variables x and y can be numbers, vectors, or matrices.
| Function | Description |
|---|---|
cos(x) |
Cosine |
sin(x) |
Sine |
tan(x) |
Tangent |
exp(x) |
Exponential |
log(x) |
Natural logarithm |
sqrt(x) |
Square root |
abs(x) |
Absolute value |
max(x) |
Maximum value |
min(x) |
Minimum value |
Apart from elementary functions, MATLAB includes predefined constant values such as pi and the imaginary unit i.
Detailed Explanation
MATLAB provides a variety of functions to handle different mathematical operations. For example, sin(x) computes the sine of x, cos(x) finds the cosine, and exp(x) gives the exponential of x. You can see that x and y, which can represent numbers or collections of numbers (like arrays), are the input values for these functions. Additionally, MATLAB has certain constants like pi, which is important in various calculations, especially those involving circles and trigonometry.
Examples & Analogies
Imagine you're a chef using a recipe. Just as a recipe tells you exactly what to do with certain ingredients at specific steps, MATLAB functions take well-defined inputs and output results based on mathematical rules.
Examples of Using Functions
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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^(-a) * sin(x) + 10βy, for a = 5, x = 2, and y = 8 is computed by:
a = 5; x = 2; y = 8; y = exp(-a)*sin(x)+10*sqrt(y) y = 28.2904
Detailed Explanation
Let's look at an example calculation in MATLAB. We define three variables: a, x, and y. We then use the exponential function exp() to compute e raised to the power of -5, and the sine function sin() to compute the sine of 2. The entire operation combines these results with additional calculations to produce a final value for y which is 28.2904. This demonstrates how you can chain functions and operations together to perform complex calculations.
Examples & Analogies
Think of it like following a multi-step mathematical operation on a recipe. Just like measuring ingredients in a recipe (you might measure flour, sugar, and butter), here, weβre using specific mathematical functions on chosen values to get our final dishβor in this case, the value of y.
Understanding Logarithmic Functions
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Note the difference between the natural logarithm log(x) and the decimal logarithm (base 10) log10(x).
To calculate sin(Ο/4) and e^10, we enter the following commands in MATLAB:
sin(pi/4) ans = 0.7071
exp(10) ans = 2.2026e+004
Detailed Explanation
In this chunk, we learn that there are different types of logarithms: the natural logarithm (log()) which uses base e, and the common logarithm (log10()) which uses base 10. Knowing this distinction is important in mathematics and engineering contexts. Additionally, we can see that when calculating sin(pi/4) in MATLAB, we get a specific output, and using exp(10) gives a very large number because it computes e raised to the power of 10.
Examples & Analogies
Imagine you are measuring distances on a map. The natural logarithm could be seen as measuring distances in a natural landscape, while the decimal logarithm measures road distancesβboth getting you somewhere different but through different paths, just like base e and base 10 logarithms lead to different results.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
-
MATLAB Functions: Essential for performing complex numerical computations efficiently.
-
Plotting Techniques: Crucial for visual comprehension of mathematical data.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
-
Calculating the value of the expression
y = exp(-5)*sin(2) + 10*sqrt(8)to illustrate the use of built-in functions. -
Creating a sine wave plot over the interval
[0, 2Ο]using the commandsx = 0:pi/100:2*piandy = sin(x).
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
π΅ Rhymes Time
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To plot a line and make it clear, label the axes and bring them near!
π Fascinating Stories
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Imagine a student plotting his points on a graph. Each point connects to show a story of mathematical beauty, just as sine waves rise and fall, dancing on the x-axis.
π§ Other Memory Gems
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Remember 'PEAK' for Plotting Essentials: Prepare your data, Execute plot, Add labels, Keep colors distinct!
π― Super Acronyms
Remember MATLAB for Mathematical Analysis, Technical computing, and Learning visualizations.
Flash Cards
Review key concepts with flashcards.
Glossary of Terms
Review the Definitions for terms.
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Term: MATLAB
Definition:
A programming environment used for numerical computing and data visualization.
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Term: Builtin Functions
Definition:
Predefined functions in MATLAB that perform standard operations.
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Term: Plotting
Definition:
The process of visualizing data in graphical form.
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Term: Vector
Definition:
An array of numbers used to represent coordinates in plotting.