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Quiz Fall 06

# Quiz Fall 06 - Michael Hansen Math 20E homework 4 These are...

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Michael Hansen Math 20E homework 4. These are the solutions of one of the two versions given. If you have the other version you should be able to use these solutions by changing some numbers at a few places. Problem 1: Consider the curve φ ( t ) = (cos( t ) , sin( t ) , 5 t ). Parametrically describe the tangent line the curve φ ( t ) at φ ( π/ 2). Answer: We find φ ( π/ 2) = ( - 1 , 0 , 5) and φ ( π/ 2) = (0 , 1 , 5 π 2 ). Thus, l ( t ) = (0 , 1 , 5 π 2 ) + ( t - π 2 )( - 1 , 0 , 5) , t R Problem 2: Determine the work done in moving along the curve φ ( t ) = (cos( t ) , sin( t ) , 5 t ) from φ (0) to φ ( π/ 2) against the force field F ( x, y, z ) = ( y, x, z ). Answer: Method 1: We use the definition of the line integral to get (recall the trigonometric identity cos 2 ( t ) - sin 2 ( t ) = cos(2 t )), φ F · d s = π/ 2 0 F ( φ ( t )) · φ ( t ) dt = π/ 2 0 (sin( t ) , cos( t ) , 5 t ) · ( - sin( t ) , cos( t ) , 5) dt = π/ 2 0 (cos 2 ( t ) - sin 2 ( t ) + 5 t ) dt = π/ 2 0 cos(2 t ) dt + π/ 2 0 5 tdt = 0 + 25 π 2 8 = 25 π 2 8 Method 2: Notice F = f where f = xy + 1 2 z 2 . i.e. F is a gradient vector field. This means the integral is independent of choice of path, and so we get (Theorem 3, p.440), φ F · d s = f

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