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Unformatted text preview: cannon (kgc299) HW01 berg (54595) 1 This printout should have 24 questions. Multiplechoice questions may continue on the next column or page find all choices before answering. 001 10.0 points Which, if any, of the following statements are true when a, b are real numbers? A. For all positive a and b , a b a + b = a b . B. For all positive a and b , a + b = radicalBig a + 2 ab + b . C. For all a and b , radicalBig ( a + b ) 2 = a + b . 1. C only 2. A only 3. B and C only 4. B only 5. none of them 6. A and C only 7. A and B only correct 8. all of them Explanation: A. TRUE: by the known difference of squares factorization, x 2 y 2 = ( x y )( x + y ) . But if a, b are positive we can set x = a and y = b . The result follows after division. B. TRUE: by the known product, ( x + y ) 2 = x 2 + 2 xy + y 2 . On the other hand, radicalBig ( x + y ) 2 =  x + y  , so if x + y > 0, x + y = radicalbig x 2 + 2 xy + y 2 . But if a, b are positive we can set x = a and y = b . The result follows since x and y are then positive. C. FALSE: radicalBig ( x + y ) 2 =  x + y  , and since radicalbig ( ) is always nonnegative, the right hand side has to be nonnegative. But if a, b can be positive or negative, an absolute value sign is then needed on the right. keywords: square root, properties of square root, PlaceUT, TrueFalse, T/F, 002 10.0 points Simplify the expression f ( x ) = 3 + 15 x 4 1 15 parenleftBig x x 2 16 parenrightBig as much as possible. 1. f ( x ) = x 4 2 x + 16 2. f ( x ) = 3( x 4) x 16 3. f ( x ) = 3( x + 4) 2 x 16 4. f ( x ) = 3( x + 4) x 16 correct 5. f ( x ) = x + 4 x + 16 cannon (kgc299) HW01 berg (54595) 2 6. f ( x ) = x 4 x + 16 Explanation: After bringing the numerator to a common denominator it becomes 3 x 12 + 15 x 4 = 3 x + 3 x 4 . Similarly, after bringing the denominator to a common denominator and factoring it be comes x 2 16 15 x x 2 16 = ( x + 1)( x 16) x 2 16 . Consequently, f ( x ) = 3 + 15 x 4 1 15 parenleftBig x x 2 16 parenrightBig = 3 x + 3 ( x + 1)( x 16) parenleftBig x 2 16 x 4 parenrightBig . On the other hand, x 2 16 = ( x + 4)( x 4) . Thus, finally, we see that f ( x ) = 3( x + 4) x 16 . 003 10.0 points Let f be the quadratic function defined by f ( x ) = 3 x 2 18 x 21 . Use completing the square to find h so that f ( x ) = 3( x h ) 2 + k for some value of k . 1. h = 9 2. h = 18 3. h = 9 4. h = 3 correct 5. h = 3 Explanation: Completing the square gives f ( x ) = 3 x 2 18 x 21 = 3( x 2 6 x 7) = 3( x 2 6 x + 9 7 9) ....
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 Spring '09
 Gualdani
 Differential Calculus

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