P9 - x 2 1 = 0 by first using differentiation to prove...

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Math 141, Problem Set #9 (due in class Mon., 11/14/11) Note: To get full credit for a problem, it is not enough to give the right answer; you must explain your reasoning. Stewart, section 3.4, problems 4, 6(ab), 8, 12, 14. Stewart, section 3.5, problems 2, 4, 6, 14, 24, 30, 38, 40. Stewart, section 3.6, problems 12, 14, 16, 22, 32, 44. Try to express your answers in the simplest possible form. For problem 22, mimic the method of Example 3 on page 184. Stewart, section 3.7, problems 2, 8, 14, 16, 30, 34, 39, 40, 49, 50. Also, do the following additional problems. A. Differentiate arctan x + arccot x and explain why your answer makes sense (in the same vein as our discussion of arccos x +arcsin x in class). B. Show that the derivative of - ln(cos x ) is tan x . C. Show that the derivative of x arctan x - 1 2 ln(1 + x 2 ) is arctan x . D. A different proof of Example 3: Show that sinh - 1 x - ln( x +
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Unformatted text preview: x 2 + 1) = 0 by first using differentiation to prove that the left-hand-side is con-stant, and then evaluating the constant by substituting a particular value for x . Tip #1: Be careful about the domain of a function and the domain of the derivative of a function. Note in particular that the latter must be a subset of the former. Tip #2: In your solutions, do not treat inf as a number. I may sometimes write this on the board as an informal shorthand, but in written solutions I don’t want to see things like arctan(inf) = π/ 2 or arctan(-inf) =-π/ 2. Please don’t forget to write down on your assignment who you worked on the assignment with (if nobody, then write “I worked alone”), and write down on your time-sheet how many minutes you spent on each problem (this doesn’t need to be exact)....
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