HW06-solutions - husain (aih243) HW06 Gilbert (56215) 1...

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Unformatted text preview: husain (aih243) HW06 Gilbert (56215) 1 This print-out should have 26 questions. Multiple-choice questions may continue on the next column or page find all choices before answering. 001 10.0 points Richard removes a turkey from the oven after it reaches a temperature of 166 F and places on a table in his dining room where the temperature is 85 F . After 10 minutes the temperature of the turkey is 158 F , while after 20 minutes it is 152 F . Use linear ap- proximation to predict the temperature of the turkey after half an hour. 1. temp 143 F 2. temp 147 F 3. temp 144 F 4. temp 146 F correct 5. temp 145 F Explanation: If T ( t ) is the temperature of the turkey after t minutes, then the given data tell us that T (0) = 166 , T (10) = 158 , T (20) = 152 , while the fact that the room temperature is 85 F means that T ( t ) gets closer to 85 F as t increases. Now by linear approximation T ( t ) T (20) + T (20)( t 20) , ( t > 20) , with T (20) = lim t 20 T ( t ) T (20) t 20 . But T (20) T (10) T (20) 10 20 = 158 152 10 = 3 5 degrees/min . Consequently, temp = T (30) 146 F . 002 10.0 points Find the linearization of f ( x ) = 1 3 + x at x = 0. 1. L ( x ) = 1 3 parenleftBig 1 + 1 6 x parenrightBig 2. L ( x ) = 1 3 parenleftBig 1 1 3 x parenrightBig 3. L ( x ) = 1 3 + 1 3 x 4. L ( x ) = 1 3 parenleftBig 1 1 6 x parenrightBig correct 5. L ( x ) = 1 3 parenleftBig 1 + 1 6 x parenrightBig 6. L ( x ) = 1 3 1 3 x Explanation: The linearization of f is the function L ( x ) = f (0) + f (0) x . But for the function f ( x ) = 1 3 + x = (3 + x ) 1 / 2 , the Chain Rule ensures that f ( x ) = 1 2 (3 + x ) 3 / 2 . Consequently, f (0) = 1 3 , f (0) = 1 6 3 , and so L ( x ) = 1 3 parenleftBig 1 1 6 x parenrightBig . husain (aih243) HW06 Gilbert (56215) 2 003 10.0 points Use linear approximation with a = 16 to estimate the number 15 . 4 as a fraction. 1. 15 . 4 3 39 40 2. 15 . 4 3 15 16 3. 15 . 4 3 77 80 4. 15 . 4 3 37 40 correct 5. 15 . 4 3 19 20 Explanation: For a general function f , its linear approxi- mation at x = a is defined by L ( x ) = f ( a ) + f ( a )( x a ) and for values of x near a f ( x ) L ( x ) = f ( a ) + f ( a )( x a ) provides a reasonable approximation for f ( x ). Now set f ( x ) = x, f ( x ) = 1 2 x . Then, if we can calculate a easily, the linear approximation a + h a + h 2 a provides a very simple method via calculus for computing a good estimate of the value of a + h for small values of h ....
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HW06-solutions - husain (aih243) HW06 Gilbert (56215) 1...

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