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一條很難的數學問題,求解答
z=f(x, y)=x^2y^3+4x-y^2+3y+5 f(2,1)=19 If f(x, y) represents the loudness (in decibels = db) at the location (x ,y) (in meters m), then what is the rate of change of the loudness at the location (2, 1) if we move in the direction <0.6, -0.8> ?
最佳解答:
dz/dt = ?z/?x.dx/dt + ?z/?y.dy/dt = 2xy^3dx/dt + 3x^2y^2dy/dt + 4dx/dt - 2ydy/dt + 3dy/dt = (2xy^3+4)dx/dt + (3x^2y^2-2y+3)dy/dt = (2*2*1^3+4)*0.6 + (3*2^2*1^2-2*1+3)*(-0.8) = 4.8 - 11.2 = -6.4 dB/s 2013-05-31 18:39:26 補充： Correction from the fourth line onwards: = 4.8 - 10.4 = -5.6 dB/s 2013-06-01 10:40:00 補充： Finding the rate of loudness change with respect to distance: Unit vector of the direction <0.6, -0.8> (u) = <0.6, -0.8>/√(0.6^2+(-0.8)^2) = <0.6, -0.8> Required rate = (?z/?x+?z/?y)?u = (2*2*1^3+4)*0.6 + (3*2^2*1^2-2*1+3)*(-0.8) = 8*0.6 + 13*(-0.8) = 4.8 - 10.4 = -5.6 dB
其他解答:
The rate of loudness change should be with respect to distance, not with respect to time. There is no indication in the question regarding time.
一條很難的數學問題,求解答
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發問:z=f(x, y)=x^2y^3+4x-y^2+3y+5 f(2,1)=19 If f(x, y) represents the loudness (in decibels = db) at the location (x ,y) (in meters m), then what is the rate of change of the loudness at the location (2, 1) if we move in the direction <0.6, -0.8> ?
最佳解答:
dz/dt = ?z/?x.dx/dt + ?z/?y.dy/dt = 2xy^3dx/dt + 3x^2y^2dy/dt + 4dx/dt - 2ydy/dt + 3dy/dt = (2xy^3+4)dx/dt + (3x^2y^2-2y+3)dy/dt = (2*2*1^3+4)*0.6 + (3*2^2*1^2-2*1+3)*(-0.8) = 4.8 - 11.2 = -6.4 dB/s 2013-05-31 18:39:26 補充： Correction from the fourth line onwards: = 4.8 - 10.4 = -5.6 dB/s 2013-06-01 10:40:00 補充： Finding the rate of loudness change with respect to distance: Unit vector of the direction <0.6, -0.8> (u) = <0.6, -0.8>/√(0.6^2+(-0.8)^2) = <0.6, -0.8> Required rate = (?z/?x+?z/?y)?u = (2*2*1^3+4)*0.6 + (3*2^2*1^2-2*1+3)*(-0.8) = 8*0.6 + 13*(-0.8) = 4.8 - 10.4 = -5.6 dB
其他解答:
The rate of loudness change should be with respect to distance, not with respect to time. There is no indication in the question regarding time.
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