Homework 8-solutions - wtm369 Homework 8 Helleloid (58250)...

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Unformatted text preview: wtm369 Homework 8 Helleloid (58250) 1 This print-out should have 22 questions. Multiple-choice 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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Homework 8-solutions - wtm369 Homework 8 Helleloid (58250)...

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