Question

A space curve Let w = x^2e^(2y) cos 3z. Find the value of dw/dt at the point (1, ln 2, 0) on the curve x = cos t, y = ln(t + 2), z = t. 

          A space curve Let w = x^2e^(2y) cos 3z. Find the value of dw/dt at the point (1, ln 2, 0) on the curve x = cos t, y = ln(t + 2), z = t.

Added by Sandra Y.

Calculus: Early Transcendentals
Calculus: Early Transcendentals
James Stewart 8th Edition
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A space curve Let w = x^2e^(2y) cos 3z. Find the value of dw/dt at the point (1, ln 2, 0) on the curve x = cos t, y = ln(t + 2), z = t.
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Transcript

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00:01 Hi, in the given problem we have space curve w is equal to x square e to the power 2 y cos 3 z now we are required to find d w over d t at 1 comma ln 2 .0 so let's replace now it is also given that x is equal to cost t y is equal to ln t plus 2 and z so, lnt plus 2 and z is equal to t.
00:40 So we can write this w as, in terms of t as, cost square t, e to the power 2, lnt plus 2 and times, times cos 3t.
01:01 So this can be further written as cost square t, t plus 2 whole square, and cost 3 t now when we differentiate so this is w now let's differentiate d w over d t so d w over d t which is to be found at 1 comma ln 2 comma 0 so this means t has to be 0 so this is when t is 0 or we can just write at t is equal to 0 we have to find these values so you can just write this as t is equal to 0 so let's differentiate and this is equal to let's differentiate first function.
01:50 This is a 2 cost t times minus sine t and the second function this is taken as constant which is t plus 2 whole square times cost t.
02:08 Next we differentiate the we'll take the first function cost square t constant and differentiate the second function using the product rule again here...
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