Question

Design and construct a computer program in Java. The following is a plot of the function f(x) = sin(x3) + x2 : In order to illustrate the effects of the two major error sources, rounding and truncation, attempt to determine an approximation to the derivative of f(x) at x = 2.0 radians using the difference approximation given below. (The true answer is 4 + 12 cos(8) or about 2.2539995942966376896). Use the formula: f'(x) ≃ (f(x+h) - f(x)) / h with h=1, 0.5, 0.25, ... 1.8189894035459e-12 (i.e., keep halving h until it is less than 2.0e-12.) Print out the values of h, your approximation to f'(x), and the error in the approximation for each value of h used. Thanks

Answer 1

Firstly, the derivative of sin(x3)+x2 is 3*x*x*cos(x)+2x or, to put it more formally:

f(x) = sin(x3)+x2

f'(x) = 3*x*x*cos(x)+2*x

you could solve sin(x) using the Taylor series for cos(x):

cos(x) = 1 - x^2/2| + x^2/4! + ...

with recursion. In Java:

class exp
{

public double cos(double x) {

  return 1 + next(-x*x/2, x, 3);

}

public double next(double term, double x, int i) {

  double next = -term * x * x / (i * (i + 1));

  return term + next(term, x, i + 2);

}

public static void main(String args[])
{
double x=2.0;
System.out.println("approximation to the derivative of f(x) at x = 2.0 “);

 System.out.println(" for f(x) = sin(x3) + x2”+ 3*x*x*cos(x)+2*x);

}
}

Error handling

For user-defined functions, when the method encounters an error during evaluation, users must use their own unchecked exceptions. The following example shows

private static class LocalException extends RuntimeException {

   // the x value that caused the problem

   private final double x;

   public LocalException(double x) {

     this.x = x;

   }

   public double getX() {

     return x;

   }

}

private static class MyFunction implements UnivariateFunction {

   public double value(double x) {

     double y = hugeFormula(x);

     if (somethingBadHappens) {

       throw new LocalException(x);

     }

     return y;

   }

}

public void compute() {

   try {

     solver.solve(maxEval, new MyFunction(a, b, c), min, max);

   } catch (LocalException le) {

     // retrieve the x value

   }

}