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Test2Review

# Test2Review - x,y 10 Calculate the directional derivative...

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Test 2 Review 1. Suppose a particle moving in three-dimensional space has position given by r ( t ) = h 2 cos t, 3 sin t, 4 t i , t 0 . Find the velocity and speed at time t = π 2 . 2. A particle is moving with velocity given by v ( t ) = h 3 2 1 + t, e - t , 1 1+ t i and initial position r (0) = h 3 , - 4 , 1 i . Find the position function r ( t ). 3. Find the length of the curve g ( t ) = ( t cos t ) i + ( t sin t ) j + ( 2 2 3 t 3 / 2 ) k , 0 t π. 4. Let f ( x, y ) = p 16 - x 2 - y 2 . Find the domain of f ( x ) and sketch the level curves for f = 0 , 1 , 2 , 3 , 4. 5. Compute the following limits: (a) lim ( x,y ) (1 , 2) x + y x - y (b) lim ( x,y ) (2 , 2) x + y x - y (c) lim ( x,y ) (0 , 0) x + y x - y (d) lim ( x,y ) (0 , 0) x 4 + y 2 x 4 - y 2 (e) lim ( x,y ) (4 , 3) x 6 = y +1 x - y + 1 x - y - 1 6. Calculate f x , f y , f xx , f xy , f yx , and f yy . (a) f ( x, y ) = e xy ln y (b) f ( x, y ) = sin 2 (3 x - 5 y ) 7. Find dw dt at t = 3 where w = x z + y z , x = cos 2 t, y = sin 2 t, z = 1 /t . 8. If f ( u, v, w ) is a differentiable function, and u = x - y, v = y - z, w = z - x , show that ∂f ∂x + ∂f ∂y + ∂f ∂z = 0 . 9. Let f ( x, y ) be a differentiable function. Using the polar coordinate transformations x = r cos θ, y = r sin θ , express ∂f ∂θ purely in terms of the rectangular coordinates (

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Unformatted text preview: x,y ). 10. Calculate the directional derivative of g ( x,y,z ) = x 2 + 2 y 2-3 z 2 at the point P (1 , 1 , 1) in the direction h 1 , 1 , 1 i . 11. Consider the curve in the plane given by the equation x 2-xy + y 2 = 7. Find the equation of the tangent line at (-1 , 2). 12. Find the equation of the tangent plane to the surface x 2-xy-y 2 = z at the point (1 , 1 ,-1). 13. Using linear approximation, estimate e . 1 sin(-. 2). 14. Find all local extrema and saddle points for the following functions (a) f ( x,y ) = x 2 + xy + 3 x + 2 y + 5 (b) f ( x,y ) = 8 x 3 + y 3 + 6 xy 15. Find the absolute maximum and minimum values of f ( x,y ) = 1+ xy-x-y on the closed region bounded by the curves y = x 2 and y = 4. 2...
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Test2Review - x,y 10 Calculate the directional derivative...

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