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Introduction to Built-in Functions
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Today, we will discuss MATLAB's built-in functions, which you will frequently use for computations. Can anyone tell me some of the basic mathematical operations we can perform?
We can calculate trigonometric values like sine and cosine!
Exactly! Functions like `sin(x)` and `cos(x)` are built into MATLAB. Remember, these are called built-ins. Another important one is the exponential function, denoted as `exp(x)`. Who can explain what this does?
It computes `e` raised to the power of `x`!
Correct! And when calculating with these, be cautious about variable assignment. Why do we need to be careful?
Because we shouldn't overwrite the built-in functions!
Right! Using names like `ii` or `jj` for loop indices can help prevent confusion. Let's move on to specific examples next.
Performing Calculations with Built-in Functions
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Let's take an example of calculating `y = exp(-a)*sin(x)+10*sqrt(y}`. If I assign `a = 5`, `x = 2`, and `y = 8`, can anyone work this out?
We need to first compute `exp(-5)` and `sin(2)`!
Exactly! After calculating these values, donβt forget to add `10 * sqrt(8}`.
I have it! The result for `y` is approximately `28.29`.
Well done! Now, let's look at logarithmic functions. Who can explain the difference between `log(x)` and `log10(x)`?
`log(x)` is the natural logarithm, and `log10(x)` is the base-10 logarithm!
Exactly! Understanding these functions is crucial because they are widely used across various fields.
Advanced Functions Example
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To exemplify further, let's calculate `sin(pi/4)` and compare it to `exp(10)`. What results do we expect?
I think `sin(pi/4)` should be around `0.7071`.
Correct! And `exp(10)` will give us a much larger number. Letβs actually run these in MATLAB.
I want to see how the outputs are different!
Absolutely! And remember, when you use commands like these, ensure your input variables are correct to avoid errors in your calculations.
We should also remember not to overwrite MATLAB's built-in functions. It can be confusing!
Great summary! Always keep your variable names distinct to keep things clear.
Introduction & Overview
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Quick Overview
Standard
The section provides hands-on examples of using MATLAB's built-in mathematical functions and calculations. Various expressions are computed, showcasing the functionality and differences between functions like natural logarithm and common logarithm, as well as basic operations involving exponential functions.
Detailed
Detailed Summary
In this section, we explore practical examples of using MATLAB's predefined mathematical functions. These functions simplify technical computing by allowing users to perform complex calculations with ease. We begin by calculating the value of mathematical expressions step-by-step, including the use of the exponential function, sine function, and logarithmic functions with specific variables assigned. For instance, we computed the expression y = exp(-a)*sin(x) + 10*sqrt(y) for given values of a, x, and y, demonstrating the application of built-in functions. Moreover, we highlight the distinctions between the natural logarithm (log) and the decimal logarithm (log10), ensuring clarity in their usage. The section also includes an example of computing sin(pi/4) and exp(10). Finally, we finish with reminders on function usage best practices, emphasizing not to reassign values to built-in functions.
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Audio Book
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Example of a Complex Expression
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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) + 10sqrt(y)
y = 28.2904
Detailed Explanation
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.
Examples & Analogies
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.
Calculating Logarithms
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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).
Detailed Explanation
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.
Examples & Analogies
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.
Sin and Exponential Calculations
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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 part, we calculate the sine of Ο/4 and the exponential function for the number 10. When we enter 'sin(pi/4)', MATLAB computes the sine of 45 degrees (or Ο/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.
Examples & Analogies
Imagine you're at a carnival. The sine function calculates how high youβd 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.
Important Notes on Built-In Functions
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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 β(-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 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.
Examples & Analogies
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.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
-
Built-in Functions: Predefined functions that simplify calculations in MATLAB.
-
Natural Logarithm vs Common Logarithm: Distinction between
log(x)andlog10(x). -
Sine Function: Calculating the sine of an angle in radians using
sin(x).
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
-
Example of calculating
exp(-5)*sin(2) + 10*sqrt(8)which evaluates to approximately 28.29. -
Calculating
log(142)yields 4.9558, whilelog10(142)gives 2.1523.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
π΅ Rhymes Time
-
To remember sin and cos real well, angles in radians are what they tell.
π Fascinating Stories
-
Imagine e as a balloon that grows exponentially, just like our math problems.
π§ Other Memory Gems
-
For logs, just think
Base is best!to remember the difference between base logarithms.
π― Super Acronyms
E.L.S. - Exponential, Logarithmic, Sine - to remember the three main functions we use.
Flash Cards
Review key concepts with flashcards.
Glossary of Terms
Review the Definitions for terms.
-
Term: Builtin Functions
Definition:
Predefined functions in MATLAB for performing mathematical calculations.
-
Term: Exponential Function
Definition:
The function
exp(x)computeseraised to the power ofx. -
Term: Logarithm
Definition:
The logarithm function, with
log(x)indicating the natural logarithm andlog10(x)for logarithm base 10. -
Term: Sine Function
Definition:
A trigonometric function to calculate the sine of an angle in radians.