2017 Fall Exam Solutions.pdf - Math 3363 Examination I Solutions Fall 2017 1 Give an example of a continuous function whose domain is an interval with

2017 Fall Exam Solutions.pdf - Math 3363 Examination I...

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Math 3363 Examination I Solutions Fall 2017 1. Give an example of a continuous function whose domain is an interval with these properties: is not the zero function on and Z ( )  = 0 for some pair of numbers and in  Note. The point of this problem is this. One way to show that a continuous function is the zero function on an in interval ( meaning ( ) = 0 for all in ) is to show that Z ( )  = 0 for each choice of and in with  However, it is not su cient to get the conclusion that you want to show Z ( )  = 0 for some choice of and . Solution. There are many correct answers. Here is one. Let = [   ] = [ 1 1] and ( ) =  The function is continuous. It is not the zero function on (for example (1 2) = 1 2 6 = 0 ), and Z 1 1 ( )  = 0 2. A rod of length (units of length), insulated except perhaps at its ends, lies along the -axis with its left end at coordinate 0 and its right end at coordinate . Let , , and be as follows. The thermal energy density (energy/length) at (units of time after the time origin) at points with fi rst coordinate is (   ) . The heat fl ux (energy/time) to the right at time through the cross section consisting of points with fi rst coordinate is (   ) . (A negative value for (   ) indicates heat fl ow to the left.) The heat energy per unit length being generated per unit time inside the rod at time at points with fi rst coordinate is (   ) . (A negative value for indicates a heat sink.) Suppose that is continuous and that and have continuous partial derivatives. Derive the equation   (   ) =   (   ) + (   ) for 0 and 0 1
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Solution. Suppose that 0 . Conservation of thermal energy tells us that the time-rate-of-change in thermal energy in the section of the rod consisting of points with fi rst coordinate satisfying is the net heat energy fl owing per unit time across the boundaries of this section plus the net heat energy being generated internally in the section. Thus  Z (   )  = (   ) (   ) + Z (   )  Since each of and have continuous fi rst order partial derivatives, we have Z   (   )  = Z  
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