handout_09_21

handout_09_21 - at 7:00 PM The following morning he starts...

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Math 1a: Intermediate Value Theorem September 21, 2009 Sample problem: Prove that there is a number c such that c 2 = 2. Sample good answer: Let f ( x ) = x 2 . We have f (0) = 0 and f (2) = 4; hence f (0) < 2 < f (2) . Since f ( x ) is continuous on [0 , 2], the intermediate value theorem says that there is a c in [0 , 2] with f ( c ) = c 2 = 2. Sample bad answer: f ( x ) = x 2 x = 0 f ( x ) = 0 x = 2 f ( x ) = 4 0 < 2 < 4 Intermediate Value Theorem. Moral: Mathematical arguments consist of complete sentences, not just formulas. 1. Is there a number x such that x 2 = cos( x )? 2. Prove that there is a number which is one less than its cube. 1
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3. A monk leaves the monastery at 7:00 AM and takes his usual path to the top of the mountain, arriving
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Unformatted text preview: at 7:00 PM. The following morning, he starts at 7:00 AM at the top and takes the same path back to the monastery, arriving at 7:00 PM. Is there a point on the path that the monk will cross at exactly the same time of day on both days? Why or why not. 4. Prove that the polynomial x 3 +2 x 2-5 x +1 has at least three zeros. (Hint: consider f (-4), f (0), f (1) and f (2).) 5. (Challenge problem) Show that at any given time there are two diametrically opposite points on the globe that have the same temperature. 2...
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This note was uploaded on 03/27/2012 for the course MATH 1a taught by Professor Benedictgross during the Fall '09 term at Harvard.

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handout_09_21 - at 7:00 PM The following morning he starts...

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