Practice exam solutions 3

Prove that the string has 0 1 stationary points in the

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Unformatted text preview: k ≥ 1. Prove that the string has 0, . − 1 stationary points in the open interval Proof The general solution with initial shape f(x) and initial velocity 0 is 2 = sin cos , = In our case, plugging in = sin you proved in HW 11 gives 2 = sin So the solution is sin and using the orthogonality of the sine functions that sin = sin = 1 = 0 ℎ cos A stationary point is a point along the string that never moves, so = 0 for all t. Taking the derivative with respect to t and setting it equal to zero gives =− sin sin So the term involving x must be zero: sin = 0 → = This gives us k-1 values in the range 0, → = = 0 for all t = 0, , . Q.E.D. , …, ,...
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