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Unformatted text preview: wtm369 Homework 8 Helleloid (58250) 1 This printout should have 22 questions. Multiplechoice questions may continue on the next column or page find all choices before answering. 001 10.0 points The table Year 1970 1980 1990 2000 Cash/capita $174 $255 $568 $920 lists the amount of U.S. cash per capita in circulation as of June 30 in the given year. Use linear approximation to estimate the amount, C (2010), of cash per capita in circu lation in the year 2010. 1. C (2010) $1273 2. C (2010) $1272 correct 3. C (2010) $1271 4. C (2010) $1274 5. C (2010) $1270 Explanation: If C ( t ) represents the cash per capita in circulation in year t and t is a continuous vari able, then the linear approximation to C ( t ) at t = 2000 is given by C ( t ) C (2000) + C (2000)( t 2000) . But in the table above C ( t ) is specified only at discrete values. To determine C (2000), therefore, we approximate it by a Newtonian Quotient: C (2000) C (2000) C (1990) 2000 1990 = 920 568 10 = 35 . 2 $/year . In this case, C ( t ) 920 + 35 . 2( t 2000) ; this provides an estimate for C ( t ) which we can assume holds for values of t close to t = 2000. Consequently, C (2010) $1272 is an estimate for the cash per capita in year 2010. 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 6 x parenrightBig correct 3. L ( x ) = 1 3 + 1 3 x 4. L ( x ) = 1 3 parenleftBig 1 1 3 x parenrightBig 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 , wtm369 Homework 8 Helleloid (58250) 2 and so L ( x ) = 1 3 parenleftBig 1 1 6 x parenrightBig . 003 10.0 points Use linear approximation with a = 9 to estimate the number 8 . 2 as a fraction. 1. 8 . 2 2 5 6 2. 8 . 2 2 4 5 3. 8 . 2 2 49 60 4. 8 . 2 2 13 15 correct 5. 8 . 2 2 17 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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This note was uploaded on 12/04/2009 for the course M 408k taught by Professor Schultz during the Fall '08 term at University of Texas.
 Fall '08
 schultz
 Calculus

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