HW2-solutions - merino (aem2588) HW2 chen (55405) 1 This...

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Unformatted text preview: merino (aem2588) HW2 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 10.0 points A machinist has to manufacture a circular metal disk with area 930 sq. cms. How close to the exact radius must the machinist control the radius if he is allowed an error tolerance of 8 sq. cms. in the area of disk? 1. within approximately 0 . 1328 cm 2. within approximately 0 . 1288 cm 3. within approximately 0 . 1318 cm 4. within approximately 0 . 1298 cm 5. within approximately 0 . 1308 cm cor- rect Explanation: The area of a circular disk of radius r is r 2 . When the area is 930 , therefore, the exact radius, r ext , of the disk is r ext = 930 . If, however, the machinist is allowed to make the disk so that its area A is within 8 of 930 then A must satisfy the inequalities 8 A 930 8 . Thus the radius r the machinist makes the disk must satisfy the inequalities 8 r 2 930 < 8 , in other words, 922 < r < 938 . Hence the area of the disk that the machinist makes will be within 8 sq. cms. of 930 sq.cms. when r ext r < 930 922 . 1314 , and r r ext < 938 930 . 1308 . Consequently, the radius r must be within approximately 0 . 1308 cms of the exact value 930. 002 10.0 points Find the value of b , b 0, for which lim x braceleftBig 5 x + b 5 x bracerightBig exists. 1. b = 27 2. b = 28 3. b = 26 4. b = 25 correct 5. b = 24 Explanation: We are told that lim x braceleftBig 5 x + b 5 x bracerightBig = A for some value A , but we arent told what the particular value of A is. The question requires us to see exactly what value b must take for A to exist. To begin, note that x + y z = x + y z x + y + z . merino (aem2588) HW2 chen (55405) 2 Thus 5 x + b 5 x = 5 x + b 25 x ( 5 x + b + 5) = 5 x x ( 5 x + b + 5) + b 25 x ( 5 x + b + 5) . Now lim x 5 x + b + 5 = b + 5 , so by properties of limits, lim x 5 x x ( 5 x + b + 5) = lim x 5 5 x + b + 5 = 5 b + 5 . Consequently, once again by the properties of limits, lim x braceleftBig b 25 x ( 5 x + b + 5) bracerightBig = lim x parenleftBig 5 x + b 5 x 5 x x ( 5 x + b + 5) parenrightBig = A 5 b + 5 . But lim x braceleftBig b 25 x ( 5 x + b + 5) bracerightBig = ( b 25) lim x braceleftBig 1 x ( 5 x + b + 5) bracerightBig . Since lim x 1 x ( 5 x + b + 5) doesnt exist, however, the only way lim x braceleftBig b 25 x ( 5 x + b + 5) bracerightBig = A 5 b + 5 can hold is if b 25 = 0 ....
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HW2-solutions - merino (aem2588) HW2 chen (55405) 1 This...

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