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# Review2 - t q for the function t = f u,v,w where u = u...

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MAC 2313-EXAM 2 Review 1. Find the length of the curve r ( t ) = 2 t 3 / 2 , cos 2 t, sin 2 t 0 t 1. 2. Find the curvature of the curve y = x 4 at the point (1 , 1). 3. Given the acceleration vector a ( t ) = t i + e t j + e - t k , v (0) = k , r (0) = j + k then find the position vector, i.e. r ( t ) = ? Also, what is the tangential component of the acceleration vector? 4. Given the vector function, r ( t ) = sin 2 t i + t j + cos 2 t k , find the unit vectors, T ( t ) =, N ( t ) =, and B ( t ) =. 5. Using the same vector function in #4, find the equation of the osculating plane and the normal plane at the point (0 , π, 1). 6. Find and sketch the domian of the function f ( x, y ) = p y - x 2 y - p 4 - x 2 - y 2 . 7. Find the limit, if it exists, or show that the limit does not exist: (a) lim ( x,y ) (0 , 0) x 6 - y 6 x 3 + y 3 (b) lim ( x,y ) (0 , 0) xy 3 x 2 + y 6 (c) lim ( x,y ) (0 , 0) 3 xy 2 x 2 + y 2 . 8. Where is the function f ( x, y ) = 3 xy 2 x 2 + y 2 if ( x, y ) 6 = (0 , 0) 1 if ( x, y ) = (0 , 0) continuous? 9. Given the function f ( s, r, t ) = t sin( r 3 s 2 t ) find the mixed partial derivative f tsr . 10. Given the function f ( x, y ) = p sin 2 x + sin 2 y + sin 2 z , find f z (0 , 0 , π/ 4). 11. Use the chain rule to write out a formula for

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Unformatted text preview: t q for the function t = f ( u,v,w ) where u = u ( p,q,r,s ) ,v = v ( p,q,r,s ) ,w = w ( p,q,r,s ) 12. Use the chain rule to nd z t where s = 2 and t = 1. z = tan u v u = 2 s + 3 t v = 3 s-2 t 13. Does there exist a function f ( x,y ) with partial derivatives f x ( x,y ) = 3 x 2 + 2 y 2 and f y ( x,y ) = 4 y-4 xy . If not, explain why. 1 14. Find the equation of the tangent plane to the ellipsoid x 2 + 4 y 2 = 169-9 z 2 at the point P(3,2,4). 15. Find the Linearization L ( x,y ) of the function f ( x,y ) = sin(2 x + 3 y ) at the point (-3 , 2). 16. Find the dierential (a) dT of the function T = v 1 + uvw and (b) dR of the function R = 2 cos . 17. Find the directional derivative of the function f ( x,y ) = ln( x 2 y ) in the direction of the vector v = &amp;lt; 4 , 3 &amp;gt; . 2...
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