a Use the mean-value theorem to show that p b - p a...

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MATH 137 Winter 2018 Assignment 4 Due: 5 pm Monday, April 2. 1. (a) Use the mean-value theorem to show that p b - p a < b - a 2 p 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 p ab is always smaller than the arithmetic mean 1 2 ( a + b ), p ab < 1 2 ( a + b ) . 2. It is di ffi cult to prove the identity arctan x + arccot x = 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 . (b) Next, show that the function f ( x ) = arctan x + arccot x has zero derivative, f 0 ( x ) = 0. (c) Use part (b) to find a suitable x that proves the identity (1). 3. Find the first three nonzero terms of the Maclaurin series for each of the following functions. a) ln 1 + x 1 - x b) sin p x p x ( x > 0) c) cosh x = e x + e - x 2 d) ln cos x Hint:

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