Calc3 Final 2016.pdf - MATH 252 Fall 2016 Practice Final 1...

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MATH 252 Fall 2016 Practice Final 1. Let a = h- 2 , 1 , 2 i and b = h- 5 , 13 , 2 i . Write b as a linear combination b = αa + c ,with ac = 0. 2. Let | u | = 6 , | v | = 8,and let θ = 3 π/ 4 be the angle between the two vectors u and v . a. Find u · v . b. Find | u × v | . 3. Find the point at which the given lines intersect: r 1 ( t ) = h 1 , 1 , 0 i + t h 1 , - 1 , 2 i r 2 ( s ) = h 2 , 0 , 2 i + s h- 1 , 1 , 0 i 4. Find the length of the curve r ( t ) = cos( t ) i + sin( t ) j + ln(cos( t )) k over the interval 0 t π/ 4. 5. Find the tangent line and the normal plane to the curve r ( t ) = h 2 t, e t , e - t i at t = 0. Equation of tangent line: Equation of normal plane: 6. Let f ( x, y, z ) = 3 xy + x 2 z and r ( t ) = h x ( t ) , y ( t ) , z ( t ) i = h e t - 1 , 3 t, t 2 i , find df/dt at t = 1. 7. If f ( x, y, z ) = x 2 - xy +3 y 2 + z 2 and ( x, y, z ) changes from (3 , - 1 , 2) to (2 . 96 , - 0 . 95 , 2 . 01), compare the values of Δ f and df . 8. Let g ( x, y, z ) = ( x + 2 y + 3 z ) 2 and P 0 = (1 , 1 , 2). At the point P 0 , find the direction v in which the function g increases the fastest and find the rate of change of g per unit distance moved in the v direction. 9a. Find the absolute minimum value of f ( x, y, z ) = 2 x 2 + 4 x + 3 y 2 - 12 y + z 2 - 2 z + 31 and the point ( x, y, z ) where this minimum is achieved. 9b. Find the absolute maximum and minimum values of f ( x, y ) = 3 x + 4 y on the unit
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  • Fall '03
  • Bologna
  • Multivariable Calculus, Coordinate system, Polar coordinate system, Joseph Louis Lagrange

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