Computer Algebra Systems Academic Semester 2011-2012 Spring Midterm Test 01 Name: ETR: e-Mail: Please send the solved worksheet in e-mail to me. Please write only into th blocks marked with "Solution". Evaluation: You can reach 100 points. -59 point: 1 60-69 point: 2 70-79 point: 3 80-89 point: 4 90- : 5

Consider the recurrence relation 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, with initial conditions 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. Find an explicit expression for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRIm5GJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn as a function of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=.

JSFH

Solve the following system of equations: 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 Determine if there are complex roots.

JSFH

This is the 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problem, or Collatz-problem. (And there are many other names.) Define in Maple the the following function on natural numbers: 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 The conjecture is that if we iterat this functions on either natural number, that is if we compute 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and so on, suddenly we com to number LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW5HRiQ2JFEjMS5GJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGLw== Try this conjecture on some mumbers.

JSFH

Compute the seventh derivative of the following expression: 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 then try to simplify the result.JSFH

JSFH

Find the minima and the maxima of the following function: 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 Plot the function and its derivative in the same coordinate system.

JSFH