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HW 1-solutions - merino(aem2588 HW 1 chen(55405 This...

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merino (aem2588) – HW 1 – chen – (55405) 1 This print-out should have 20 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0points Evaluate lim x 2 (3 x 2 + 3 x 6) . Correct answer: 12. Explanation: Since a polynomial is continuous, lim x 2 (3 x 2 + 3 x 6) = (3 x 2 + 3 x 6) vextendsingle vextendsingle vextendsingle x =2 = 12 . 002 10.0points If a ball is thrown into the air with a velocity of 60 ft/sec, its height in feet after t seconds is given by y = 60 t 16 t 2 . Find the average velocity of the ball for the time period beginning at t = 1 and lasting 1 / 4 seconds. 1. average vel. = 24 ft/sec correct 2. average vel. = 23 ft/sec 3. average vel. = 25 ft/sec 4. average vel. = 22 ft/sec 5. average vel. = 26 ft/sec Explanation: If the height of the ball after t seconds is given by y = y ( t ) = 60 t 16 t 2 , then the average velocity of the ball over a time interval [1 , t + h ] is avg. vel. = y (1 + h ) y (1) h . Now y (1 + h ) = 60(1 + h ) 16(1 + h ) 2 = 60(1 + h ) 16(1 + 2 h + h 2 ) = 44 + 28 h 16 h 2 , while y (1) = 60 16 = 44 . Thus y (1 + h ) y (1) h = 28 h 16 h 2 h = 28 16 h . Consequently, when h = 1 / 4, therefore, average velocity = 24 ft/sec . 003 10.0points Evaluate lim x 2 (4 x 2 + 3 x 2) . 1. limit = 18 2. limit = 20 correct 3. limit = 19 4. limit doesn’t exist 5. limit = 22 Explanation: By properties of limits, the limit can be taken term by term in f ( x ) = 4 x 2 + 3 x 2 . Consequently, lim x 2 (4 x 2 + 3 x 2) = 20 .
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merino (aem2588) – HW 1 – chen – (55405) 2 004 10.0points Determine which of the following functions is even. 1. f ( x ) = x 3 7 x 2. f ( x ) = 9 x - 2 correct 3. f ( x ) = 5 x - 3 4. f ( x ) = 9 x 3 + 5 x 2 + 4 5. f ( x ) = x 5 5 x 3 Explanation: 005 10.0points Find the composition f g of the functions f ( x ) = 2 x + 4 x 2 , g ( x ) = 3 x + 1 . 1. ( f g )( x ) = 6 x 2 + 14 x + 4 x 2 2. ( f g )( x ) = 6 x + 6 x 2 3. ( f g )( x ) = 2 x + 4 9 x 2 + 6 x + 1 4. ( f g )( x ) = 6 x + 6 9 x 2 + 6 x + 1 correct 5. ( f g )( x ) = 4 x 2 + 16 x + 16 x 2 Explanation: The composition f g of functions f, g is defined by ( f g )( x ) = f ( g ( x )) . When f ( x ) = 2 x + 4 x 2 , g ( x ) = 3 x + 1 , therefore, ( f g )( x ) = 2 g ( x ) + 4 g ( x ) 2 = 2(3 x + 1) + 4 (3 x + 1) 2 = 6 x + 6 9 x 2 + 6 x + 1 . 006 10.0points A glass window is designed in the shape of a rectangle surmounted by a semi-circle as shown in the figure below h 7 The width of the window is 7 feet, but the height h is to be determined. If the area of the window is 84 sq. ft., find what the height has to be. (Use π 22 / 7). 1. height = 12 . 25 ft 2. height = 13 . 5 ft 3. height = 12 . 5 ft 4. height = 13 . 25 ft 5. height = 12 . 75 ft correct Explanation: As the figure h - r r 7
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merino (aem2588) – HW 1 – chen – (55405) 3 shows, the radius r of the semi-circular area is 7 / 2. Thus the total area of the window is the sum 7 parenleftbigg h
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