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Midterm2-math108-Williams

Midterm2-math108-Williams - Math 108 Williams Midterm 2 60...

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Math 108 Williams Midterm 2 - 4/14/2010 60 Minutes; no notes, books, calculators; you may use the official formula sheets. Name: ______________________________________________________ Section 03 / Section 04 To receive credit, you must show your work, explain your reasoning, and demonstrate the use of the specific method indicated, when one is indicated. Clarity will be considered in grading. “I have adhered to the Duke Community Standard in completing this examination.” Signature: ___________________________________ 1. _____ 2. _____ 3. _____ 4. _____ 5. _____ Total (out of 24): ______
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1 1. (3 points) Find the eigenfunction associated with eigenvalue 𝜆 0 = 0 , or prove that 𝜆 0 = 0 is not an eigenvalue: ? ′′ + 𝜆? = 0, 0 < ? < 1; ? 0 + ? 0 = 0, ? 1 = 0. 2. (3 points) Suppose 𝜓 1 , 𝜓 2 , is an orthonormal basis of real valued functions on the interval 0 < ? < 1 . If ? ? 𝜓 ? ? =1 converges to ? ( ? ) and ? ? 𝜓 ? ? =1 converges to ? ( ? ) , prove that ? , ? = ? 1 ? 1 + ? 2 ? 2 + .
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2 3. (8 points) Consider the following heat conduction problem: ? ? = ? ?? + 25, 0 < ? < 1, ? > 0 ? ? 0, ? = ? ? 1, ? = 0 ? ? , 0 = cos 𝜋? a. Sketch ? ( ? , 𝑇 ) for a large, fixed value 𝑇 . Label your sketch clearly. b. Solve.
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3 4. (5 points) Consider the temperature of a wire that loses heat to the surrounding
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