Math 233 Warm Up Problems September 15, 2009 1. Find the gradients - PowerPoint PPT Presentation

Math 233 Warm Up Problems September 15, 2009

1. Find the gradients of the functions (a) x f ( x , y ) = ∇ f = x + y 2 (b) x f ( x , y , z ) = ∇ f = y + z 2 2. Supose you have a function f ( x , y ) and you know that ∂ f ∂ f f (4 , 7) = 10 ∂ x (4 , 7) = 3 ∂ x (4 , 7) = − 2 Determine your best estimates for the following f (5 , 7) ≈ f (3 , 7) ≈ f (4 , 8) ≈ f (4 , 6) ≈

1. Find the gradients of the functions (a) y 2 � 2 xy � x f ( x , y ) = ∇ f = ( y 2 + x ) 2 , − ( y 2 + x ) 2 x + y 2 (b) x � 1 x 2 xz � f ( x , y , z ) = ∇ f = z 2 + y , − ( z 2 + y ) 2 , − ( z 2 + y ) 2 y + z 2 2. Supose you have a function f ( x , y ) and you know that ∂ f ∂ f f (4 , 7) = 10 ∂ x (4 , 7) = 3 ∂ x (4 , 7) = − 2 Determine your best estimates for the following f (5 , 7) ≈ 13 f (3 , 7) ≈ 7 f (4 , 8) ≈ 8 f (4 , 6) ≈ 12

Lecture Problems 3. f ( x , y ) = x 2 y 3 f xx = f xy = f yx = f yy = 4. f ( x , y ) = x sin( y ) D 1 D 1 f = D 2 D 1 f = D 1 D 2 f = D 2 D 2 f =

Lecture Problems 3. f ( x , y ) = x 2 y 3 f xx = 2 y 3 f xy = 6 xy 2 f yx = 6 xy 2 f yy = 6 xy 4. f ( x , y ) = x sin( y ) D 1 D 1 f = 0 D 2 D 1 f = cos y D 1 D 2 f = cos y D 2 D 2 f = − x sin y

5. f ( x , y ) = cos( xy ) ∂ 2 f ∂ 2 f ∂ x 2 f = ∂ y ∂ x f = ∂ 2 f ∂ 2 f ∂ x ∂ y f = ∂ y 2 f = 6. Let f ( x , y , z ) = x + y 2 + z 3 r ( t ) =(sin t , cos t , t ) ( f ◦ r ) ′ ( t ) =

5. f ( x , y ) = cos( xy ) ∂ 2 f ∂ 2 f ∂ x 2 f = − y 2 cos xy ∂ y ∂ x f = − sin xy − xy cos xy ∂ 2 f ∂ 2 f ∂ y 2 f = − x 2 cos xy ∂ x ∂ y f = − sin xy − xy cos xy 6. Let f ( x , y , z ) = x + y 2 + z 3 r ( t ) =(sin t , cos t , t ) ( f ◦ r ) ′ ( t ) = − 2 cos t sin t + cos t + 3 t 2