undefined Solution The values of t which corresponds to the maxima and minima of the oscillation of y ( t ) = e − t sin t ; check the answer by graphing y ( t ) = e − t sin t . The values of t which corresponds to the maxima and minima of the oscillation of y ( t ) = e − t sin t is t = π 4 , and t = 5 π 4 _ , respectively. It is given that, y ( t ) = e − t sin t (1) Differentiate (1) with respect to t as follows. y ′ ( t ) = e − t ( − 1 ) ⋅ sin t + e − t cos t ( 1 ) = − e − t sin t + e − t cos t Substitute y ′ ( t ) = 0 in the above equation and simplify further. − e − t sin t + e − t cos t = 0 − e − t sin t = − e − t cos t e − t sin t = e − t cos t sin t cos t = e − t e − t tan t = 1 It is known that, tan t = 1 for t = π 4 + n π where n = 0 , 1 , 2 , ... . That implies, for n = 0, t = π 4 That is, for n = 0, the value of t = π 4 . For n = 1, t = π 4 + π = π + 4 π 4 = 5 π 4 That is, for n = 1, the value of t = 5 π 4 . From the above calculation, it is observed that the local maxima occurs at t = π 4 and local minima occurs at t = 5 π 4 . Thus, the values of t which corresponds to the maxima and minima of the oscillation of y ( t ) = e − t sin t is t = π 4 , and t = 5 π 4 _ , respectively. Use online graphing calculator and sketch the graph of y ( t ) = e − t sin t and check the point where the maxima and minima of the oscillation of y ( t ) occurs as shown in below Figure 1. From Figure 1, it is observed that the local maximum is at t = π 4 and local minimum is at t = 5 π 4 .

10th Edition

ISBN: 9781119664697

Author: Kreyszig

Publisher: WILEY

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