solutions_09_21

# solutions_09_21 - time of day on both days? Why or why not....

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Math 1a: Intermediate Value Theorem (Solutions) September 21, 2009 1. Is there a number x such that x 2 = cos( x )? Solution. Yes, there is. Consider the function f ( x ) = x 2 - cos( x ). We have f (0) = - 1 and f ( π/ 2) = ( π/ 2) 2 ; hence f (0) < 0 < f ( π/ 2). Since the function is continuous on the interval [0 ,π/ 2], the intermediate value theorem says that there is an x in [0 ,π/ 2] such that f ( x ) = 0, or equivalently x 2 = cos( x ). 2. Prove that there is a number which is one less than its cube. Proof. We want to show that there is an x such that x = x 3 - 1. Consider f ( x ) = x - x 3 . Then f (0) = 0 and f (2) = - 6. So, f (2) < - 1 < f (0). Since - 1 is between f (0) and f (2) and f is continuous on [0 , 2], we use the intermediate value theorem to conclude that there is an x such that f ( x ) = - 1. In other words, x - x 3 = - 1; that is x = x 3 - 1. 3. A monk leaves the monastery at 7:00 AM and takes his usual path to the top of the mountain, arriving 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
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Unformatted text preview: time of day on both days? Why or why not. Solution. This one is fun to do, so I wont give the full solution. A hint is to consider the following functions f ( x ) = The monks altitude at time x on day 1 , g ( x ) = The monks altitude at time x on day 2 Then f and g are continuous functions on the interval [7 am , 7 pm]. We want to show that f ( x ) = g ( x ) for some x in [7 am , 7 pm]. Equivalently, we want to show that f ( x )-g ( x ) = 0. Now consider the value of f ( x )-g ( x ) for x = 7 am and x = 7 pm and see what you get. 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).) Solution. Use the hint. That is, compute f (-4), f (0), f (1) and f (2) and see what you get. 5. (Challenge problem) Show that at any given time there are two diametrically opposite points on the globe that have the same temperature. 1...
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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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