exam1.sol - x the inequality ( x 2-4) / ( x 2-9) > 0 is...

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EXAM 1 SOLUTIONS – MATH 12, FALL 2011 Solution 1. The area of the rectangle is ‘w , the perimeter is 2 + 2 w . If the area is 10 then = 10 /w , hence the perimeter is 2(10 /w ) + 2 w . Solution 2. The difference quotient of a function f is one of f ( x ) - f ( a ) x - a or f ( x + h ) - f ( x ) h . If f ( x ) = x 2 - 5 x then the quotient ( f ( x ) - f ( a )) / ( x - a ) is x 2 - 5 x - a 2 + 5 a x - a = x 2 - a 2 x - a + 5 a - 5 x x - a = ( x + a ) - 5 . Solution 3. We want to know the largest possible domain of the function f ( x ) = 1 2 - x + 1. It is visibly all real numbers except 2. To graph this we graph, in order, the functions 1 /x, 1 / ( x - 2) , 1 / ( x - 2) - 1 - (1 / ( x - 2) - 1) = 1 / (2 - x ) + 1 . Solution 4. The vertex of the parabola described by f ( x ) = - 5 x 2 + 5 x + 13 is determined by the formula ( - b/ (2 a ) ,f ( - b/ (2 a )). - b 2 a = - 5 2 · ( - 5) = 1 / 2 = f (1 / 2) = - 5 / 4 + 5 / 2 + 13 = 57 / 4 . This parabola is opening downwards, hence the function f ( x ) has a maximum. The maximum value of f is the second coordinate of its vertex, which is 57 / 4. The distance of the vertex (1 / 2 , 57 / 4) to the point (1 , 1) is p (1 - 1 / 2) 2 + (1 - 57 / 4) 2 . Solution 5. We want to know for which
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Unformatted text preview: x the inequality ( x 2-4) / ( x 2-9) > 0 is valid. We factor the numerator and denominator to get ( x-2)( x + 2) ( x-3)( x-3) > . We know that for large values of x , each factor ( x ± 2), ( x ± 3) is positive. Further, each factor changes sign exactly once and this happens when the factor is zero. Thus, the inequality is true for large x and changes • from true to false at x = 3, • from false to true at x = 2, • from true to false at x =-2, • from false to true at =-3. 1 2 EXAM 1 SOLUTIONS – MATH 12, FALL 2011 Thus the inequality is true if x is in (-∞ ,-3) ∪ (-2 , 2) ∪ (3 , ∞ ) Solution 6. The equation of a circle with center (3 ,-4) and radius 5 is ( x-3) 2 + ( y-(-4)) 2 = 5 2 , ( x-3) 2 + ( y + 4) 2 = 25 ....
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This note was uploaded on 11/13/2011 for the course MATH 12 taught by Professor Marx during the Fall '08 term at UC Davis.

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exam1.sol - x the inequality ( x 2-4) / ( x 2-9) > 0 is...

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