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# P8 - x and cos x but there are inﬁnitely many others Show...

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Math 141, Problem Set #8 (due in class Mon., 11/7/11) Note: To get full credit for a non-routine problem, it is not enough to give the right answer; you must explain your reasoning. Stewart, section 3.2, problems 6, 8, 10, 16, 18, 22, 24, 30, 34, 40, 44, 46, 52, 60, 64, 72, 76. For problem 34, note that a picture of the graph of a function is not by itself a proof that the function is one-to-one; also keep in mind that the stated domain of f is { x R : x > 1 } . Stewart, section 3.3, problems 8, 14, 16, 30, 38, 42, 44, 50, 54, 56, 58, 60(ab), 64, 68. Also: A. Find the inverse function of f ( x ) = 3 x + | x | . (For partial credit, express it as a piecewise-linear function; for full credit, express it via a single formula involving the absolute value function.) B. Prove: If f is odd and one-to-one then f - 1 is odd. C. Discuss the limit lim x π + ln(sin x ) (compare with problem 72 from sec- tion 3.2). D. Suppose the function f has the property that f ( x +2 π ) = f ( x ) for all x . (Examples of such functins are sin
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Unformatted text preview: x and cos x , but there are inﬁnitely many others.) Show that the function f also has this property. That is, show that f ( x +2 π ) = f ( x ) for all x . (This homework problem has nothing to do with inverse functions etc.; it’s just more practice in the art of proving things using deﬁnitions and known theorems.) Important note : For this assignment, and future assignments, avoid short-hands like “lim x → q 1 /x = √ + ∞ = + ∞ ” that pretend that + ∞ and-∞ are numbers. Instead, say things like “As x → ∞ , 1 /x becomes arbitrarily large and hence so does its square root.” 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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