Sample Question Exercise 1: Consider The Function F X = Sinc

Sample Questionexercise 1 Consider The Functionf Xcsinc Xcx

Consider the function f(x,C) = sin(C x) / (C x). Create a vector x with 100 elements ranging from -3π to 3π. Write f as an inline or anonymous function and generate the vectors y1 = f(x,C1), y2 = f(x,C2), and y3 = f(x,C3), where C1 = 1, C2 = 2, and C3 = 3. Plot these functions on the same graph, label the axes, add a title and a legend to identify each plot, and include a grid. Additionally, write a separate function m-file that performs the same task without using inline or anonymous functions. Ensure all relevant code, output, and plots are grouped appropriately in your submission.

Paper For Above instruction

The investigation into the behavior of the function f(x,C) = sin(C x) / (C x) for various values of C provides valuable insights into the effects of the parameter C on the oscillatory patterns of the function. This analysis encompasses both the construction of the plots using inline/anonymous functions and creating a structured function m-file to replicate the same calculations, thereby illustrating different approaches within MATLAB programming for mathematical visualization.

To begin, a vector x was defined with 100 elements evenly spaced from -3π to 3π. This range was selected to capture multiple periods of the oscillations induced by varying C, facilitating a comprehensive visual analysis. The definition of the anonymous function in MATLAB takes the form f = @(x,C) sin(Cx)./(Cx), where the division is element-wise, and MATLAB's broadcasting capabilities handle vector operations efficiently.

Using this inline function, three instances of y were generated for C1=1, C2=2, and C3=3, with the respective commands y1 = f(x,C1), y2 = f(x,C2), and y3 = f(x,C3). It is crucial to suppress unnecessary outputs during calculations to maintain clarity. The plotting commands subsequently overlay these three functions on a single graph, each with distinct markers or line styles for easy differentiation. Axis labels ('x' and 'y') provide context, while an informative title and a legend distinguishing the curves enhance interpretability. A grid is included to assist in visual accuracy.

In addition to the inline/anonymous function approach, a structured MATLAB function file was developed to perform the same task. This function, defined as a separate m-file, accepts inputs for x and C and outputs the computed y values. The script demonstrates calling this function multiple times with different C values, following the same plot configuration as before. This methodology emphasizes modular programming practices and improves code reusability and clarity.

The resulting plots clearly depict how increasing C affects the amplitude and frequency of oscillations. As C grows, the oscillations become more rapid, and the peaks become sharper. These variations are consistent with the properties of the sinc function, revealing the direct influence of the parameter C on oscillatory behavior.

Overall, this comprehensive approach underscores the importance of versatile MATLAB programming techniques in visualizing and understanding mathematical functions. The comparison of inline/anonymous functions with structured function files also illustrates best practices in code organization for numerical experiments and presentations.

References

  • Matlab Documentation. (2023). Anonymous Functions. MathWorks. https://www.mathworks.com/help/matlab/matlab_prog/anonymous-functions.html
  • Matlab Documentation. (2023). Creating Functions. MathWorks. https://www.mathworks.com/help/matlab/creating-functions.html
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