PS3_answers

PS3_answers - Math 237 A1. dw (2) dt Problem Set 3 Answers...

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Math 237 Problem Set 3 Answers A1. dw dt ( - 2) = 76. A2. a) If f , x ( t ), y ( t ) are differentiable then g 0 ( t ) = f x ( x ( t ) ,y ( t )) x 0 ( t ) + f y ( x ( t ) ,y ( t )) y 0 ( t ) . b) If f is differentiable at (1 , 0) we get g 0 (0) = 4. A3. If f is differentiable at (1 , 1 , 1) then we get g 0 (1) = - 13 A4. If f is differentiable then g (1 , 2) = (14 , - 8). A5. i) ∂w ∂t = ∂w ∂x ∂x ∂t + ∂w ∂y ∂y ∂t + ∂w ∂z ∂z ∂t . ii) dz dt = f x ( x,y ) + f y ( x,y ) g 0 ( x ) iii) ∂z ∂u = f x ( x,y ) h u ( u,v ) + f y ( x,y ) g 0 ( x ) h u ( u,v ) iv) dw dz = f x ( x,y,z ) + f y ( x,y,z )( g x ( x,z ) + g z ( x,z ) h 0 ( x )) + f z ( x,y,z ) h 0 ( x ) v) ± ∂w ∂p ² q = ± ∂F ∂p ² q,r,s + ∂F ∂r ∂r ∂p + ∂F ∂s ∂s ∂p . A7. a) (2 e 2 t cos t - e 2 t sin t, 2 e 2 t sin t + e 2 t cos t, 2), b) - 13 30 A8. i) a) 1 5 , b) 1 in the direction ± 1 2 , - 1 2 ² . ii) a) ( π 4 + 1 ) 2 4 - π 8 , b) q ( π 4 ) 2 + 1 2 in the direction ± π 4 2 , π 4 2 , 1 2 ² . A9. D ( ˆ i + ˆ j ) T (0 ,π/ 4) = 0. Greatest rate of change is 10 in the direction ± - 10 2 , - 10 2 ² . A10. 1 3 A11. 5. A12. y - 1 = - ( x - 2) A13. f x ( a,b,c )( x - a
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