midterm2_practice_wi11

# midterm2_practice_wi11 - x 2 y 2 x = e u v y = u v 12 Find...

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Practice Problems for Exam 2 1) A bullet is shot, from the origin, at an angle of 60 , with respect to the positive x -axis, with initial speed v 0 = 75 ft/s. Find the position vector for the bullet. 2) Find the length of the path (3 t 2 , 4 t 3 ) over the interval 1 t 4. 3) Find the speed of r ( t ) = h sin 3 t, cos 4 t, cos 5 t i at t = π/ 2. 4) Find the velocity vector v ( t ) given the accelaration vector h e t , 0 , t + 1 i and the initial velocity v (0) = h 1 , - 3 , 2 i . 5) Describe the horizontal and vertical traces of f ( x, y ) = x 2 + 4 y 2 . 6) Draw a contour map of f ( x, y ) = x + y 2 . Show four level curves. 7) Find the limit if it exists. If the limit does not exist, prove that it does not exist. (a) lim ( x,y ) (0 , 0) 7 x 2 - 5 y 2 x 2 + 1 (b) lim ( x,y ) (0 , 0) x ( y 2 + 1) y 8) Find z x and z y if z ( x, y ) = 3 xe x 2 - y 2 . 9) Find the equation of the tangent plane for f ( x, y ) = x y at (4 , 4). Use it to estimate f (3 . 99 , 4 . 01). 10) Find the directional derivative of f ( x, y ) = x 2 y 3 at P = ( - 2 , 1) in the direction v = ~ i + ~ j . 11) Find ∂f ∂u
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Unformatted text preview: x 2 + y 2 , x = e u + v , y = u + v . 12) Find the critical points of f ( x,y ) = x 3-xy + y 3 . Use the second derivative test to classify them. 13) Use Lagrange Multipliers to ﬁnd the extreme values of f ( x,y ) = xy subject to 4 x 2 + 9 y 2 = 32. Solutions 1) r ( t ) = h 75 2 t,-16 t 2 + 75 √ 3 2 t i 2) 1 2 (65 3 / 2-5 3 / 2 ) 3) 5 4) r ( t ) = h e t ,-3 , 1 2 t 2 + t + √ 2 i 5) Horizontal Traces: Ellipses, Vertical Traces: Parabolas 7) (a) 0 (b) d.n.e. 8) z x = 3 e x 2-y 2 + 3 xe x 2-y 2 (2 x ), z y = 3 xe x 2-y 2 (-2 y ) 9) z = 1 2 x-1 4 y + 1, f (3 . 99 , 4 . 01) ≈ 1 . 9925 10) 4 √ 2 11) 2( e u + v ) e u + v + 2( u + v ) 12) (0 , 0) saddle, (1 / 3 , 1 / 3) local min. 13) 8 / 3 is max.,-8 / 3 is min. 1...
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