1. Suppose z = x2 sin y, x = -5s2 + 2t2, y = -10st. Use the chain rule to find a
ID: 3342799 • Letter: 1
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1.
Suppose z = x2 sin y, x = -5s2 + 2t2, y = -10st. Use the chain rule to find and as functions of x, y, s and t. = -20xssiny - 10tx^2cosy = 8xtsiny-10sx^2cosy Find the numerical values of and when (s, t) = (3, 4). (3, 4) = 2704.50 (3, 4) = 2895.27 Suppose w = x/y + y/z, where x = et, y = 2 + sin (2t), and z = 2 + cos (5t). Use the chain rule to find dw/dt as a function of x, y, z, and t. Do not rewrite x, y, and z in terms of t, and do not rewrite et as x. dw/dt = exp(t)/y + 2cos(2t)/z - 2xcos(2t) + 5ysin(5t)/z^2 Note: You may want to use exp() for the exponential function. Your answer should be an expression in x, y, z, and t; e.g. "3x - 4y" Use part A to evaluate dw/dt when t = 0. 1/y - 2/z - 2x/y^2Explanation / Answer
a ) dZ /ds = dz / dx * dx / ds + dz / dy * dy /ds
= 2x sin y * -25s + x^2cosy * -10t = -50 sx siny - 10x^2tcosy
b ) dZ /dt = dz / dx * dx / dt + dz / dy * dy /dt
= 2x sin y * 4t + x^2cosy * -10s = 8 tx siny - 10x^2scosy
a ) dw/dt = dw / dx * dx /dt + dw / dy * dy /dt + dw / dz * dz /dt
= 1/y * e^t + (-x/y^2 + 1/z) * (2cos2t) + (-y /z^2) * (-5sin5t)
= e^t /y -2xcos2t /y^2 + 2cos2t /z + 5ysin5t /z^2
b ) at t = 0 , dw/dt = 1/y - 2x/y^2 + 2/z
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