Multivariable Calculus Staple your work and use this page as a cover page. Find
ID: 2880274 • Letter: M
Question
Multivariable Calculus Staple your work and use this page as a cover page. Find dz/dt if z = x^2 + y^2 + xy, x = sin t, and y = e^t. Find dw/dt if w =- xe^y/z, x = t^2, y = -t, and z = 1 + 2t. If z = f(x, y) and f is differentiable with x = g(t) and y = h(t), use the following table of values to compute dz/dt at t = 3. g(3) = 2 g'(3) = 5 f_x (2, 7) = 6 h(3) = 7 h'(3) = -4 f_y (2, 7) = -8 Find partial differential w/partial differential r and partial differential w/partial differential theta at r = 2 and theta = pi/2 if w = xy + yz + zx, x = cos theta, t = r sin theta, and z = r theta.Explanation / Answer
(1)
x = sin t, so dx/dt = cos t
y = e^t, so dy/dt = e^t
z = x^2 + y^2 + xy
dz/dt = 2x(dx/dt) + 2y(dy/dt) + x(dy/dt) + y(dx/dt)
= 2 sin t cos t + 2 e^t e^t + (sin t)(e^t) + (e^t)cos t
= 2 sin t cos t + 2 e^(2t) + (e^t)(sin t + cos t)
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