1.8. Exercises

Determine the result of the following calculations using SCILAB if a=2.3, b=-2.3, c=π/2, x=2/π, and y=√3:

1. (a² + bc + x)
2. sin(c) + y/c
3. (a + c)/(x + y)
4. 1/(cos(c) + ln(x))
5. (a + c)³/b

In this exercise, students will practice using SCILAB to perform mathematical operations. The formulae provided involve the use of SCILAB to compute values based on the given variables: a, b, c, x, and y. To get the results, they can enter each calculation line by line in the SCILAB environment. For example:

1. For the first calculation, they would input 'result1 = a^2 + b*c + x' to get the answer.
2. For the second, they would calculate 'result2 = sin(c) + y/c'. Each calculation utilizes basic arithmetic, trigonometric functions, and logarithms, showcasing the mathematical capabilities of SCILAB.

Think of this exercise like a cooking recipe where specific ingredients (the variables a, b, c, x, and y) are mixed in different combinations (the calculations). Just as you would follow a recipe step by step to create a dish, you're following the mathematical steps to arrive at a final answer using SCILAB.

Check if the following Boolean statements are true or false based on the values of a, b, c, x, and y given above:

6. a > c
7. a = b
8. (2a + b)/x² < 1
9. x + 2ab + b² ≤ 23
10. 2ac = 2cb

This part of the exercise challenges students to evaluate Boolean expressions using SCILAB. Each expression can be input directly into SCILAB to see if it results in 'true' or 'false'. For instance:

• To check 'a > c', you would type in 'a > c' and SCILAB will return either '1' (true) or '0' (false) based on the condition.
• Students will learn how different arithmetic operations and comparisons work in SCILAB, enhancing their understanding of logical reasoning and evaluation.

Consider these Boolean statements like a series of gatekeeping questions before entering a club. Each statement is a question that determines whether the condition is satisfied (true), allowing entry, or not (false), denying entry. Just like a doorman checks IDs and other requirements, SCILAB checks if the statements hold true based on the input values.

11. Use SCILAB’s help facility to find out information about function deff and use it to define a function y = f(x) = x² + 1.

In this task, students are introduced to defining functions in SCILAB. They are first encouraged to explore the built-in help facility to understand how to craft new functions using 'deff'. After they familiarize themselves with this command, they write their own function to calculate y values based on a given x input. The learning here is about function creation, crucial for creating reusable code in programming.

Think of this exercise as creating a new recipe that can be reused whenever you want to cook a dish. When you define a function in SCILAB, you're essentially creating a recipe that takes specific ingredients (inputs) and processes them to produce a dish (output), making cooking (programming) more efficient and enjoyable.

Using the vectors u = [3, 2, -1], and v = [4, -6, 2], calculate the following operations:

12. w = u + v
13. r = u ./ v
14. z = v * u
15. t = v .* u

This section focuses on performing operations on vectors, which is a fundamental aspect of SCILAB. Students will learn to combine operations such as addition and multiplication between two vectors. For instance:

• To compute 'w = u + v', students will learn how each corresponding element is added together. Using './' allows for element-wise division, and '*' denotes matrix multiplication. The differentiation helps them understand both element-wise and matrix operations, critical when working with linear algebra.

Imagine u and v as two different teams racing towards a finish line. Each team member has their own speed (vector elements), and the results of operations like addition or multiplication can be visualized as how their combined efforts impact the race's outcome. Just like how you would analyze team strategies to win a race, you'll analyze how vectors interact mathematically in SCILAB.

Using the matrices A, B, and C shown below, perform the following operations:

A = | 1 -2 0 | | 4 -1 -2 | B = | -1 -1 10 | | 0 4 -2 | C = | 2 5 | | 4 2 |

16. A + B
17. A ⋅ B
18. B ⋅ A
19. B ⋅ C
20. A ⋅ B ⋅ C

Here, students engage with matrix operations which are key in various applications, from solving systems of equations to transformations in computer graphics. Through operations such as addition and multiplication, students learn to manipulate matrix elements. They discover how different types of multiplications (dot product vs. element-wise) work, expanding their understanding of linear algebra concepts in SCILAB.

Imagine matrices as large spreadsheets where each cell contains data. Just like you can sum up columns or multiply cells together according to certain rules, matrices in SCILAB can be manipulated to explore data relationships and solutions to problems. This resembles how data analysts aggregate and transform data to derive insights in real-world applications.

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