137_PS5_Optimization_SOLUTIONS.pdf - Problem Sets u2013...

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Problem Sets – Mean Value Theorem and Optimization These problem sets contain additional practice problems that correspond to the lecture material. There are hundreds more in the calculus textbooks on reserve at the library, or on the Davis shelves close to QA303. Mean Value Theorem 1. Prove that, for k > 0, 1 k + 1 < ln(1 + k ) - ln k < 1 k This set of inequalities was used in Problem Set 1 (Question 8) to prove that the sequence a n = 1+ 1 2 + 1 3 + · · · + 1 n - ln n has a limit. 2. 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 ) .
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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