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Exercises

 

 

 

 

 

 

Exercise 1

 

Given a right triangle with a base = 8 and a height = 4. Assume that the units are time and velocity respectively.

 

  Using an appropriate function and limits of integration, use Trapeze, Simpson, and Romberg integration to find the enclosed area with a precision of 4 decimal
    places.

 

  Can it be said that the area represents a meaningful value (I would consider the product of ants and algorithms a meaningless value)? If so, what is it?

 

Exercise 2

 

  Use the first 5 terms of the Taylor series expansion to find the area under the exponential (ex) from 2 to 10


Exercise 3

 

Evaluate the area under the curve of f(x) = x2 + 7 from x = -1 to x = 5

Compare the answer provided by the 3 algorithms (Newton, Simpson, and Romberg) for the following precisions and iterations

 

    Precision    Iterations

    10E-5         10    20    50

    10E-6         10    20    50    

    10E-7         10    20    50          

    10E-8         10    20    50

 

Exercise 4

 

Integrate f(x) = x from 2 to 6 using Newton, Simpson, and Romberg for each of the precisions listed for Exercise 3. Record the number of iterations required to attain that precision, along with the answers. Compare the answer with the exact answer and list  the delta. Do the results agree with what you expected?