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Calc III Homework 5 1. Express dw/dt as a function of t by using Chain Rule (a) w = x z + yz , with x = cos t + sin t, y = cos t − sin t and z = 1/t (b) w = z − sin xy, with x = t, y = ln t and z = et−1 ∂z ∂z 2. Compute ∂u and ∂v where z = arctan (x/y) with x = u cos v and y = ∂z ∂z u sin v. Evaluate ∂u and ∂v when (u, v) = (−2, 0) . 3. Let w = x2 e2y cos 3z. Find the value of dw/dt at the point (1, ln 2, 0) on the curve c(t) = cos t, ln t2 + 2 , t , t ∈ R. 4. Let T = g(x, y) = xy − 2 be the temperature at the point (x, y) on the ellipse. Find the maximum and minimum values of T if √ √ x = 2 2 cos t, y = 2 sin t, 0 ≤ t ≤ 2π 5. Find ∇f at the given point f (x, y, z) = x2 + y 2 − 2z 2 + z ln x at (1, 1, 1) 6. Find the derivative of the function g(x, y) = → − → − → − u = 12 i + 5 j . x−y xy+2 in the direction of 7. Find the derivative of the function h(x, y) = cos(xyz) + ln(zx) in the → − → − → − − direction of → u =2 i + j −2k. 8. Find the points on the ellipsoid x2 + 2y 2 + 3z 2 = 1 where the tangent plane is parallel to the plane 3x − y + 3z = 1. Hint: think about normal vectors 9. Find the points on the surface z 2 = xy + 1 which are closest to the origin. Hint: you want to minimize the distance away from the origin of points in R3 with the constraint function g(x, y, z) = z 2 − xy − 1 10. Find the dimensions of the rectangular box with largest volume if the total surface area is given as 100cm2 . 11. Use Lagrange multiplier to minimize the function f (x, y, z) = yz + xy subjected to the conditions xy = 1 and y 2 + z 2 = 1. 12. Find the local max, local min and saddle points to the functions if they exist (a) f (x, y) = x2 − y 2 (b) f (x, y) = ex cos y 1