1. A bar with initial temperature profile f (x) > 0, with ends held at 0...

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1. A bar with initial temperature profile f (x) > 0, with ends held at 0° C, will cool as t→∞, and approach a steady-state temperature 0°C. However, whether or not all parts of the bar start cooling initially depends on the shape of the initial temperature profile. The following example may enable you to discover the relationship. (a) Find an initial temperature profile f (r), 0≤ x ≤ 1, which is a linear combination of sin tx and sin 37x, and satisfies f(0) = 0 = 4(1), f (¹) 2. = 1. A bar with initial temperature profile f (x) > 0, with ends held at 0° C, will cool as t→∞, and approach a steady-state temperature 0°C. However, whether or not all parts of the bar start cooling initially depends on the shape of the initial temperature profile. The following example may enable you to discover the relationship. (a) Find an initial temperature profile f (r), 0≤ x ≤ 1, which is a linear combination of sin tx and sin 37x, and satisfies f(0) = 0 = 4(1), f (¹) 2. =

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