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Answer Guide 1 Math 408C: Unique Numbers 56975, 56980, and 56985 Tuesday, September 1, 2009 Homework problems Section 1.1 22. A spherical balloon with radius r + 1 has volume V ( r + 1) = 4 3 ( r + 1) 3 = 4 3 ( r 3 + 3 r 2 + 3 r + 1) : r to radius r + 1 is V ( r + 1) V ( r ) = 4 3 (3 r 2 + 3 r + 1) : ( F ) 24. Because f ( x ) = x 3 , we have f ( a + h ) = a 3 + 3 a 2 h + 3 ah 2 + h 3 : Therefore, f ( a + h ) f ( a ) h = 3 a 2 h + 3 ah 2 + h 3 h = 3 a 2 + 3 ah + h 2 : ( F ) 26. We compute that f ( x ) f (1) x 1 = x +3 x +1 4 2 x 1 = ( x + 3) 2( x + 1) ( x 1)( x + 1) = x + 1 ( x 1)( x + 1) = 1 x + 1 : ( F ) 36. Notice that H ( t ) = 4 t 2 2 t = (2 + t )(2 t ) 2 t : The domain of H is f t : t 6 = 2 g . As long as t 6 = 2 , we can cancel, obtaining H ( t ) = 2 + t . The graph y = H ( t ) is the same as the graph y = 2 + t with the point 2 ; 4) removed. 54.

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