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Unformatted text preview: Math 107 Third Midterm Examination November 22, 2005 NAME (Please print) Page Score 2 3 4 5 6 7 8 9 10 Total (Max Possible: 100) Instructions: 1. Do all computations on the examination paper. You may use the backs of the pages if necessary. 2. Put answers inside the boxes (when applicable). 3. Please signify your adherence to the honor code: I, , have neither given nor received aid in completion of this examination. 1 (10 Points) Score Given that F [ a, b ] is the set of all functions on the closed set [ a, b ] and F [ a, b ] is a linear space, determine whether each of the following sets of functions is a subspace of F [ a, b ] . 1. (2 points) All functions f ∈ F [ a, b ] for which f ( a ) = 0 . Subspace. If f 1 ( a ) = 0 = f 2 ( a ) then f 1 ( a ) + f 2 ( a ) = 0 and cf 1 ( a ) = 0, so the set is closed under element addition and scalar multiplication. 2. (3 points) All functions f ∈ F [ a, b ] for which f ( a ) = 1 . Not a subspace. If f 1 ( a ) = 1 = f 2 ( a ) then f 1 ( a ) + f 2 ( a ) = 2, so the set is not closed under element addition. 3. (2 points) All continuous functions f ∈ F [ a, b ] for which R b a f ( x ) dx = 0 . Subspace. If R b a f 1 ( x ) dx = 0 = R b a f 2 ( x ) dx then R b a f 1 ( x ) + f 2 ( x ) dx = 0 and R b a cf 1 ( x ) dx = 0, so the set is closed under element addition and scalar multiplication. 4. (3 points) All differentiable functions f ∈ F [ a, b ] for which f ( x ) = f ( x ) . Subspace. If f 1 ( x ) = f 1 ( x ) and f 2 ( x ) = f 2 ( x ) then ( f 1 + f 2 ) ( x ) = f 1 ( x ) + f 2 ( x ) and ( cf 1 ) ( x ) = cf 1 ( x ), so the set is closed under element addition and scalar multiplication. 2 (15 Points) Score A can of orange juice concentrate is taken from a freezer at 5 ◦ into a room with temperature 20 ◦ degrees. After 30 minutes, the temperature of the orange juice con centrate is 2 ◦ degrees. Newton’s Law of Cooling states that the rate of change of the temperature of an object is proportional to the difference between its own temperature and the ambient temperature (i.e. the temperature of its surroundings). 1. (5 points) Describe the variables you will use to solve this problem, including their units. Let t be time in minutes, T ( t ) be temperature in degrees and k be the proportionality constant in Newton’s law, with units of 1 /minutes . 2. (5 points) Describe your model for this problem, including a differential equation and one or more conditions on the solution. Newton’s law says dT dt ( t ) = k [ T ( t ) 20]; the initial condition is T (0) = 5 and the other condition is T (30) = 2....
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This note was uploaded on 08/03/2009 for the course MATH 107 taught by Professor Trangenstein during the Spring '07 term at Duke.
 Spring '07
 Trangenstein
 Math

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