Exercise PrintTriangles: Write a method to print each of the following patterns using nested-loops in a class called PrintTriangles. The program shall prompt user for the numRows. The signatures of the methods are: public static void printXxxTriangle(int numRows)  // Xxx is the pattern's name                             1                         1   2   1                     1   2   4   2   1                 1   2   4   8   4   2   1             1   2   4   8  16   8   4   2   1         1   2   4   8  16  32  16   8   4   2   1     1   2   4   8  16  32  64  32  16   8   4   2   1 1   2   4   8  16  32  64 128  64  32  16   8   4   2   1                   (a) PowerOf2Triangle

Exercise PrintTriangles: Write a method to print each of the following patterns using nested-loops in a class called PrintTriangles. The program shall prompt user for the numRows. The signatures of the methods are:

public static void printXxxTriangle(int numRows)  // Xxx is the pattern's name

                            1

                        1   2   1

                    1   2   4   2   1

                1   2   4   8   4   2   1

            1   2   4   8  16   8   4   2   1

        1   2   4   8  16  32  16   8   4   2   1

    1   2   4   8  16  32  64  32  16   8   4   2   1

1   2   4   8  16  32  64 128  64  32  16   8   4   2   1

                  (a) PowerOf2Triangle

1                                      1

1  1                                 1   1

1  2  1                            1   2   1

1  3  3  1                       1   3   3   1

1  4  6  4  1                  1   4   6   4   1

1  5 10 10  5  1             1   5  10  10   5   1

1  6 15 20 15  6  1        1   6  15  20  15   6   1

(b) PascalTriangle1           (c) PascalTriangle2

Exercise TrigonometricSeries: Write a method to compute sin(x) and cos(x) using the following series expansion, in a class called TrigonometricSeries. The signatures of the methods are:

public static double sin(double x, int numTerms)   // x in radians

public static double cos(double x, int numTerms)

Compare the values computed using the series with the JDK methods Math.sin(), Math.cos() at x=0, π/6, π/4, π/3, π/2 using various numbers of terms.

Hints: Do not use int to compute the factorial; as factorial of 13 is outside the int range. Avoid generating large numerator and denominator. Use double to compute the terms as:

Exercise Exponential Series: Write a method to compute e and exp(x) using the following series expansion, in a class called TrigonometricSeries. The signatures of the methods are:

public static double exp(int numTerms)   // x in radians

public static double exp(double x, int numTerms)

Exercise SpecialSeries: Write a method to compute the sum of the series in a class called SpecialSeries. The signature of the method is:

public static double sumOfSeries(double x, int numTerms)

Exercise FibonacciInt (Overflow) : Write a program called FibonacciInt to list all the Fibonacci numbers, which can be expressed as an int (i.e., 32-bit signed integer in the range of [-2147483648, 2147483647]). The output shall look like:

F(0) = 1

F(1) = 1

F(2) = 2

...

F(45) = 1836311903

F(46) is out of the range of int

Hints: The maximum and minimum values of a 32-bit int are kept in constants Integer.MAX_VALUE and Integer.MIN_VALUE, respectively. Try these statements:

System.out.println(Integer.MAX_VALUE);

System.out.println(Integer.MIN_VALUE);

System.out.println(Integer.MAX_VALUE + 1);

Take note that in the third statement, Java Runtime does not flag out an overflow error, but silently wraps the number around. Hence, you cannot use F(n-1) + F(n-2) > Integer.MAX_VALUE to check for overflow. Instead, overflow occurs for F(n) if (Integer.MAX_VALUE – F(n-1)) < F(n-2) (i.e., no room for the next Fibonacci number).

Write a similar program for Tribonacci numbers.

Exercise FactorialInt (Overflow): Write a program called Factorial1to10, to compute the factorial of n, for 1≤n≤10. Your output shall look like:

The factorial of 1 is 1

The factorial of 2 is 2

...

The factorial of 10 is 3628800

Modify your program (called FactorialInt), to list all the factorials, that can be expressed as an int (i.e., 32-bit signed integer in the range of [-2147483648, 2147483647]). Your output shall look like:

The factorial of 1 is 1

The factorial of 2 is 2

...

The factorial of 12 is 479001600

The factorial of 13 is out of range

Hints: The maximum and minimum values of a 32-bit int are kept in constants Integer.MAX_VALUE and Integer.MIN_VALUE, respectively. Try these statements:

System.out.println(Integer.MAX_VALUE);

System.out.println(Integer.MIN_VALUE);

System.out.println(Integer.MAX_VALUE + 1);

Take note that in the third statement, Java Runtime does not flag out an overflow error, but silently wraps the number around.

Hence, you cannot use F(n) * (n+1) > Integer.MAX_VALUE to check for overflow. Instead, overflow occurs for F(n+1) if (Integer.MAX_VALUE / Factorial(n)) < (n+1), i.e., no room for the next number.

Try: Modify your program again (called FactorialLong) to list all the factorial that can be expressed as a long (64-bit signed integer). The maximum value for long is kept in a constant called Long.MAX_VALUE.