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Math 320 (Fall 2005) Second Test ______________________ Calculus III name (8 points)
This enjoyable 50 minute test covers Covers Sections 12.7,13.1-4,14.1-2 of Calculus by Stewart 5th ed. Relax and do well. Unless otherwise stated, problems are 4 points.
- Evaluate the integral ⌡
⎮
⌠
⎜
⎛ ⎠
⎟
1+ t^2 i^ +^
2 t 1+ t^2 k^ dt
- Write the following equations in the coordinate system stated.
a) z =
2 xy x^2 + y^2 in cylindrical coordinates
b) x^2 + y^2 + z^2 = 2 xz
- Sketch the curve r ( t ) = < sin t , cos t , 3 > and indicate the direction in which t increases.
- A particle moves with the position function r ( t ) = cos t i + sin t j + t k. Find the following.
velocity v ( t )
speed v(t)
acceleration a ( t )
unit tangent vector T ( t )
the derivative T’ ( t )
unit normal vector N (0) (note t = 0)
binormal vector B (0)
- Find the domain of the function of f( x,y,z ) =
x^2 + y^2 − 4 ln( x^2 + y^2 − 9)
- Find the following limits or show that they do not exist.
a.
lim ( x,y,z )→(1,−1,0)
x^2 y + xy^2 − xyz x + y − z
b.
lim ( x,y )→(0,0)
3 x^2 y x^4 + y^2
- Find the length of the curve r =
9 (1+ t )
3/2 (^) i +^4 9 (1− t )
3/2 (^) j +^1 3 t^ k^ from^ t^ = 0 to^ t^ = 4. (8 points)
