137_PS5_Optimization.pdf - Problem Sets u2013 Mean Value...

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Problem Sets – Mean Value Theorem and OptimizationThese problem sets contain additional practice problems that correspond to the lecture material. There are hundredsmore in the calculus textbooks on reserve at the library, or on the Davis shelves close to QA303.Mean Value Theorem1. Prove that, fork >0,1k+ 1<ln(1 +k)-lnk <1kThis set of inequalities was used in Problem Set 1 (Question 8) to prove that the sequencean= 1+12+13+· · ·+1n-lnnhas a limit.2. a) Use the mean-value theorem to show thatb-a <b-a2afor 0< a < b.b) Use the result from part (a) to show that for two positive numbers 0< a < b, the geometric meanabisalways smaller than the arithmetic mean12(a+b),ab <12(a+b).3. It is difficult to prove the identityarctanx+ arccotx=π2,(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, arccotx, is,ddxarccotx=-11 +x2.Hint:The cotangent cotx= cosx/sinx, and obeys the identity 1 + cot2x= csc2x.

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