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M408K Hm. 1 Solutions

M408K Hm. 1 Solutions - Mahon Kevin Homework 1 Due 3:00 am...

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Mahon, Kevin – Homework 1 – Due: Sep 13 2005, 3:00 am – Inst: Edward Odell 1 This print-out should have 25 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. The due time is Central time. 001 (part 1 of 1) 10 points Simplify the difference quotient f ( x + h ) - f ( x ) h , ( h 6 = 0) , as much as possible when f ( x ) = 4 x 2 + 7 x - 16 . 1. 8 x + 7 2. 8 x - 7 + 4 h 3. 8 x + 7 + 4 h correct 4. 4 x - 7 + 4 h 5. 4 x + 7 + 4 h Explanation: Now f ( x + h ) = 4( x + h ) 2 + 7( x + h ) - 16 = 4 x 2 + 8 xh + 7 x + 4 h 2 + 7 h - 16 , while f ( x ) = 4 x 2 + 7 x - 16 . Thus f ( x + h ) - f ( x ) = 8 xh + 4 h 2 + 7 h . Consequently, f ( x + h ) - f ( x ) h = 8 x + 7 + 4 h . 002 (part 1 of 2) 10 points There is a linear relationship T F = mT C + b between the temperature T F on the Fahren- heit scale and its equivalent T C on the Centi- grade scale. As the thermometers 32 212 0 100 Freezing Boiling Farenheit Centigrade show, water freezes at 32 F and boils at 212 F, whereas it freezes at 0 C and boils at 100 C. Determine the relationship between T c and T F 1. T C = 9 5 T F + 32 2. T F = 9 5 T C - 32 3. T F = 5 9 T C + 32 4. T F = 9 5 T C + 32 correct 5. T F = 5 9 T C - 32 Explanation: Since water freezes at 32 F and at 0 C, T C = 0 = T F = 32 in the linear equation T F = mT C + b , so b = 32. On the other hand, since water boils at 212 F and at 100 C, T C = 100 = T F = 212; i.e. , 212 = 100 m + 32, so m = 9 5 . Conse- quently, T F = 9 5 T C + 32 .
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Mahon, Kevin – Homework 1 – Due: Sep 13 2005, 3:00 am – Inst: Edward Odell 2 003 (part 2 of 2) 10 points Convert - 35 2 C to its Fahrenheit equivalent. 1. - 35 2 C 1 2 F correct 2. - 35 2 C ∼ - 1 2 F 3. - 35 2 C ∼ - 5 2 F 4. - 35 2 C ∼ - 3 2 F 5. - 35 2 C 3 2 F Explanation: Substituting for T C = - 35 2 in T F = 9 5 T C + 32 , we see that - 35 2 C is equivalent to - 63 2 + 32 F. Hence - 35 2 C 1 2 F. 004 (part 1 of 1) 10 points Find the function g that results after the following transformations are applied in the order given to the graph of a function f : (1) stretch along y -axis by a factor 8; (2) reflect about the x -axis; (3) shift down 3 units; (4) reflect about the y -axis. 1. g ( x ) = 24 - 8 f ( x ) 2. g ( x ) = 24 + 8 f ( - x ) 3. g ( x ) = 8 f ( x ) - 24 4. g ( x ) = - 3 - 8 f ( - x ) correct 5. g ( x ) = 24 - 3 f ( - x ) Explanation: Beginning with a function y = f ( x ), the four transformations change f ( x ) as follows: f ( x ) (1) -→ 8 f ( x ) (2) -→ - 8 f ( x ) (3) -→ - 3 - 8 f ( x ) (4) -→ - 3 - 8 f ( - x ) . Consequently, g ( x ) = - 3 - 8 f ( - x ) . 005 (part 1 of 1) 10 points If f ( x ) = x 2 , which one of the following is the graph of the function g defined by g ( x ) = - f ( x - 1) + 2 ? 1. -4 -3 -2 -1 1 2 -1 1 2 3 4 5 6 7 2. -1 1 2 3 4 5 -1 1 2 3 4 5 6 7
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Mahon, Kevin – Homework 1 – Due: Sep 13 2005, 3:00 am – Inst: Edward Odell 3 3. -2 -1 1 2 3 4 -3 -2 -1 1 2 3 correct 4. -2 -1 1 2 3 4 -1 1 2 3 4 5 6 7 5. -2 -1 1 2 3 4 -1 1 2 3 4 5 Explanation: The graph of f ( x ) = x 2 is a parabola open- ing upwards and passing through the origin. To obtain the graph of the function g ( x ) = - f ( x - 1) + 2 from the graph of the function f ( x ) = x 2 , therefore, we perform the following transfor- mations on the graph of f : Shift it 1 unit(s) to the right; reflect the obtained graph over the x -axis; finally, shift the graph 2 units up.
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