137_Assignment4_SOLUTIONS.pdf - MATH 137 Winter 2018 u2013...

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MATH 137 Winter 2018 – Assignment 4 SOLUTIONS 1. (a) Use the mean-value theorem to show that b - a < b - a 2 a for 0 < a < b .
(b) Use the result from part (a) to show that for two positive numbers 0 < a < b , the geometric mean ab is always smaller than the arithmetic mean 1 2 ( a + b ), ab < 1 2 ( a + b ) .
2. It is difficult to prove the identity arctan x + arccot x = π 2 , ( x > 0) (1) directly, but we can exploit the properties of the derivative to facilitate the proof. (a) First, show that the derivative of the inverse cotangent, arccot x , is, d dx arccot x = - 1 1 + x 2 . Hint: The cotangent cot x = cos x/ sin x , and obeys the identity 1 + cot 2 x = csc 2 x .

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