Elaborating some earlier programs
        
        
            - 
                New version of the table of factorials program:  write
                to a file instead of the screen.
            
 
            - 
                New version of the program to compute the integral of
                the function f(x) = fourth root of 1 - x^2 on the
                interval [0,1]:  it should declare f as a C function.
            
 
            - 
                The integral of e^{x^2/2}
            
 
            - 
                The series 1 + 1/2^3 + 1/3^3 + 1/4^3 + ...
            
 
        
        
        
            Problem 1.
        
        Rewrite your factorial program so that it writes its output to a
        file using fopen and fprintf.
        
        
            Problem 2
        
        Rewrite your program to compute the integral of the fourth root
        of 1 - x^2 on the interval [0,1] so that it uses a function
        declaration.
        
        
            Problem 3
        
        Modify the previous program to compute the integral of
        e^{-x^2/2} on the interval [-1,1].  Aim for two decimals of
        accuracy and discuss the precision actually obtained.  Make a
        sketch of the geometric figure whose area the integral
        represents.
        
        
            Problem 4
        
        Write a program which computes the sum
  f(1) + f(2) + \cdots + f(n)
        where f is an arbitrary function delared as a C function and
        where the user specifies n.  Use this program to compute an
        approximation to the sum
   beta = 1 + 1/2^3 + 1/3^3 + 1/4^3 + ...
        accurate to two decimal places.  What value of n need to achieve
        this accuracy?  Justify your answer.
        
        
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