Problem 1 :
If
u(x, y) = x2y+3xy4, x = et and y = sin t
find du/dt and evaluate it at t = 0.
Problem 2 :
If
u(x, y, z) = xy2z3, x = sint, y = cost, z = 1+e2t
find du/dt Solution
Problem 3 :
If
w(x, y, z) = x2+y2+z2, x = et, y = et sint and z = etcost
find dw/dt. Solution
Problem 4 :
Let
U(x, y, z) = xyz, x = e-t, y = e-tcost, z = sint, t ∈ℝ.
Find dU/dt. Solution
Problem 5 :
If
w(x, y) = 6x3-3xy+2y2, x = es, y = cos s, s ∈ ℝ
find dw/ds, and evaluate at s = 0. Solution
Problem 6 :
If
z(x,y) = x tan-1(xy), x = t2, y = set, s,t ∈ ℝ.
Find ∂z/∂s and ∂z/∂t at s = t = 1 Solution
Problem 7 :
Let
U(x, y) = ex sin y
where x = st2, y = s2t s,t ∈ ℝ. Find ∂U/∂s, ∂U/∂t and evaluate them at s = t = 1. Solution
Problem 8 :
Let
z(x, y) = x3-3x2y3
where x = set, y = se-t, s, t ∈ ℝ. Find ∂z/∂s, ∂z/∂t.
Problem 9 :
W(x, y, z) = xy + yz + zx, x = u − v, y = uv, z = u + v, u,v ∈ ℝ. Find ∂W/∂u, ∂W/∂v and evaluate them at (1/2, 1).
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