sols7 - SOLUTIONS TO ASSIGNMENT #7 1. Find the...

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SOLUTIONS TO ASSIGNMENT #7 1. Find the linearizations L ( x ) of the following functions f ( x )near x =0 . (a) f ( x )= 25 + x 2 + x . (b) f ( x )=(1 - 2 x ) β , where β is some constant. (c) f ( x )=ln( x + 1 - x 2 ) . Solution: In all cases the linearization near x =0is L ( x f (0) + f 0 (0) x. (a) f 0 (0) = 2 x +1 2 25 + x 2 + x ± ± ± x =0 = 1 10 and f (0) = 5 = L ( x )=5+ 1 10 x. (b f 0 (0) = - 2 β (1 - 2 x ) β - 1 ± ± ± x =0 = - 2 β and f (0)=1= L ( x )=1 - 2 βx. (c) f 0 (0) = 1+ 1 2 (1 - x 2 ) - 1 / 2 ( - 2 x ) x + 1 - x 2 ! ± ± ± x =0 =1and f (0)=0= L ( x x. 2. Two towns on the trans Canada Hiway are 100 km apart. Two cars leave the first town at 1:00 pm and both arrive at the second town one hour later. Show that at sometime between 1:00 pm and 2:00 pm they had the same velocity. Solution: Let the positions of the 2 cars be x 1 ( t )and x 2 ( t ) respectively, and set x ( t x 1 ( t ) - x 2 ( t ) . Then x (0) = 0 and x (1)=0 , and so by the Mean Value Theorem there exists t such that 0 <t< 1and x 0 ( t )=0 . Therefore the cars have the same speed at this time. 3. Find the length of the longest ladder that can be carried horizontally around a corner, from a corridor am wide to one that is bm wide.
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sols7 - SOLUTIONS TO ASSIGNMENT #7 1. Find the...

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