EXERCISE 8.6
1. If u ( x, y ) = x2y + 3xy4 , x = et and y = sin t, find du/dt and evaluate it at t = 0 .
2. If u ( x, y , z) = xy2z3 , x = sin t , y = cos t , z = 1+ e2t , find du/ dt .
3. If w( x, y , z) = x2 + y2 + z2 , x = et , y = et sin t and z = et cos t , find dw/ dt .
4. Let U ( x, y , z) = xyz, x = e−t , y = e−t cos t , z = sin t , t ∈ ℝ. Find dU/dt .
5. If w( x, y ) = 6x3 − 3xy + 2y2 , x = es , y = cos s , s ∈ ℝ, find dw/ds , and evaluate at s = 0 ,
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.
7. Let U ( x, y ) = exsin y , where x = st2 , y = s2t, s , t ∈ ℝ. Find ∂U/∂s , ∂U/∂t and evaluate them at s = t = 1.
8. Let z ( x, y ) = x3 − 3x2y3 , where x = set , y = se−t , s , t ∈ ℝ. Find ∂z/∂s and ∂z/∂t.
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).
Answers:
1. et ( 2et sin t + 3sin4 t + et cos t + 12 sin3 t cos t ) , 1
2. (1 + e2t )2 [cos3 t (1+ e2t ) − sin t sin 2t (1+ e2t ) + 6e2t sin t cos2 t]
3. 4e2t
4. − e−2t [sin 2t − cos 2t ]
5.18e3s − 3es cos s + 3es sin s − 4 sin s cos s , 15
8. 3s3 (e3t + s2 e−t ), 3s2 et (e2t −5e−2t s2 )
9. 2u (1+ 2v ) , 2 (u2 – v) , 3, −3/2
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