**Unformatted text preview: **For f ( x ) = x n , f ( x ) = nx n-1 . (You may use the product rule for derivatives ( fg ) = f g + fg in your argument.) (3) Prove that the sum of the interior angles of any convex n-gon equals ( n-2)180 ◦ . (4) 1.4.7: Guess a formula for A n and prove it using induction. (5) Let A 1 ,A 2 ,...,A n be nonsingular m × m matrices. Prove that the product A 1 A 2 ··· A n is nonsingular and that ( A 1 ··· A n )-1 = A-1 n ··· A-1 1 . (*) Bonus: Prove that the Principle of Mathematical Induction and the Well-Ordering Principle (stated below) are logically equivalent. Well-Ordering Principle: Every non-empty set of positive integers has a smallest element; i.e. for any non-empty set of natural numbers S , there exists an m ∈ S such that m ≤ s for all s ∈ S . 1...

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