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This exercise involves running and completing MATLAB programming tasks based on provided examples. The tasks include understanding variable assignment, evaluating expressions, calculating geometric parameters, plotting functions, input/output operations, and implementing numerical methods such as the Bisection method. You will analyze example MATLAB scripts, then adapt and write your own scripts to perform similar computations and visualizations, culminating in solving for a root of a function using the Bisection method.

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MATLAB is a versatile programming environment widely used in engineering, mathematics, and scientific computing for its powerful computational and visualization capabilities. This exercise introduces foundational MATLAB concepts, including variable assignment, expression evaluation, plotting functions, input/output handling, and implementing numerical algorithms. Working through these tasks enhances understanding of MATLAB syntax, programming logic, and application to real-world problems.

Understanding Variable Assignment and Expressions

One fundamental aspect of MATLAB programming is assigning values to variables. For example, the statement z = 10 creates a variable named z, stores the value 10 in it, and allocates space in memory known as the workspace. This allows for subsequent calculations or manipulations using z. Evaluating expressions involves computational operations such as exponentiation, multiplication, addition, and division. For example, the problem x = 3; y = 4; followed by (x^2*y^3)/(x-y)^2 calculates and outputs the value resulting from these operations.

In MATLAB, the command ans displays the output of the last executed expression, storing the result in a default variable. To practice this, students evaluate a more complex expression with specific variable values, such as x=2 and y=-1, with the goal of computing arithmetic involving these variables using MATLAB syntax.

Calculating Geometric Quantities

MATLAB simplifies calculating geometric parameters such as the area of a circle or volume of a cylinder. For example, to compute the area of a circle with radius r=2.5, the script calculates area = pi * r^2. Displaying messages and results can be achieved with the disp function combined with string concatenation using square brackets [] and num2str to convert numerical values to strings.

Similarly, calculating the volume of a cylinder involves multiplying the base area by its height (length). The MATLAB program prompts the user or directly assigns the dimensions, then computes and displays the area and volume messages. This approach demonstrates how MATLAB can be used to handle both static data and dynamic user inputs, facilitating versatile computational tasks.

Function Plotting

MATLAB's plotting capabilities allow visualizing mathematical functions. For instance, to plot sin(x) over a specified interval, an array of x-values is generated, and the corresponding y-values are computed and plotted. The function sin(x) generates the sine wave, which can be visualized with labels and titles. Plotting helps in understanding the behavior of functions and analyzing their properties visually.

Further, students are tasked with plotting the absolute value of sin(t) over a symmetric interval from -2π to 2π, with a specified increment. The MATLAB function abs() computes absolute values, and plotting these enhances understanding of absolute functions and their symmetry.

Input and Output in MATLAB

Interactive MATLAB scripts can prompt users for information using the input command. For example, a script may ask for a user's name and then display a personalized message. The format 's' specifies string input, enabling collection of textual data. Concatenating strings and variables with [ ] constructs custom messages for display with disp.

Building on this, students will write a script that prompts for a name and address, then displays this data in a mailing address format, suitable for printing on an envelope. These practices demonstrate MATLAB's ability to handle user data, string processing, and output formatting.

Numerical Methods: The Bisection Technique

The final challenge involves applying the Bisection method to find a root of the function f(x) = x^3 - 9. The Bisection method narrows the interval containing the root by iteratively evaluating the function at the midpoint and selecting subintervals where the sign change occurs. A sensible initial interval must be chosen, such as an interval where the function changes sign, e.g., between 2 and 3, since f(2) = -1 and f(3) = 18.

Implementing this method in MATLAB involves coding a loop that computes the midpoint, evaluates the function, checks the sign, and updates the interval until the approximation reaches an acceptable error threshold or a maximum number of iterations. This numerical approach emphasizes MATLAB’s capability in solving nonlinear equations efficiently and precisely, critical in engineering and scientific analyses.

Conclusion

This exercise underscores MATLAB’s strengths in programming for mathematical, geometrical, and data processing tasks. By working through variable assignments, expression evaluations, plotting, input/output operations, and implementing numerical algorithms, students deepen their proficiency in MATLAB. Mastery of these fundamental techniques serves as a foundation for advanced applications across scientific and engineering domains.

References

  • Matlab Documentation. (2023). MATLAB User Guide. The MathWorks, Inc.
  • Chapra, S. C., & Canale, R. P. (2015). Numerical Methods for Engineers. McGraw-Hill Education.
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  • Kiusalaas, J. (2013). Numerical Methods in Engineering with MATLAB. Cambridge University Press.
  • Matlab Central. (2023). MATLAB File Exchange. Retrieved from https://www.mathworks.com/matlabcentral
  • Stewart, J. (2016). Calculus: Early Transcendentals. Cengage Learning.
  • Roache, P. J. (1994). Computational Fluid Dynamics. Hermosa Publishers.
  • Dennis, J. E., & Schnabel, R. B. (1996). Numerical Methods for Unconstrained Optimization and Nonlinear Equations. SIAM.
  • Press, W. H., Teukolsky, S. A., Vetterling, W. T., & Flannery, B. P. (2007). Numerical Recipes: The Art of Scientific Computing. Cambridge University Press.
  • Gander, M. J. (2004). Elsevier’s Encyclopedia of Mathematics. Elsevier Science.

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