Name TEST 2 No Calculators 1 16 points A particle s acceleration at time t is given by d2 r t sin t cos t cos t sin t 0 dt2 dr initial velocity 0 1 1 1 initial position r 0 0 0 0 dt Find the particle s position r t at time t and the arc length of its trajectory from time t 0 to t 1 Integration gives velocity r0 cos t sin t sin t cos t 0 c and c needs to be such that the initial velocity is correct Hence r0 cos t sin t sin t cos t 1 Integration gives position r sin t cos t cos t sin t t c1 and c1 needs to be such that the initial position is correct Hence r t sin t cos t 1 1 cos t sin t t Since r0 2 cos t sin t 2 sin t cos t 2 1 3 we have that the arc length equals Z 1 r0 t dt 3 0 2 14 points Does f x y xy x y have a limit as x y approaches 0 0 Justify your answer No Solving f x y 1 gives y x 1 x hence f x x 1 x 1 as x 0 Solving f x y 1 gives y x 1 x hence f x x 1 x 1 as x 0 By the Two Path Test the limit does not exist 3 14 points Find the value of z x at the point 1 1 1 if the equation xy z 3 x 2yz 0 defines z as a function of the two independent variables x and y and the partial derivative exists Differentiation gives y 3z 2 zx x z 3 2yzx 0 hence zx 2 4 14 points Let w f r where r and are the polar coordinates i e x r cos and y r sin Express wx as a function of r and r x2 y 2 tan 1 y x rx x r cos x y x2 y 2 sin r wx fr rx f x fr cos f sin r p 1 2 3 4 5 6 7 8 5 14 points Find the equations of tangent plane and of the normal line to the surface z x2 y 2 at the point 2 1 3 f x2 y 2 z 0 fx 2x 4 fy 2y 2 fz 1 f 4 2 1 Tangent plane 4 x 2 2 y 1 z 3 0 Normal line x 2 4t y 1 2t z 3 t 6 14 points Let f x y z x y yz Give a good estimate of the maximum increase of f as we move a distance 0 01 from the point 1 1 1 fx 1 y 1 fy x y 2 z 0 fz 1 f 1 0 1 f 2 f increases the most in the direction u f f df f u f ds u df f ds 0 01 2 7 14 points Let f x y e 1 1 x2 1 y cos x2 y 3 Find the linearization of f at the point x2 2x xy2 1 1 e cos x2 y 3 e y sin x2 y 3 2x 2 y x2 x2 x2 1 1 fy 2 e y cos x2 y 3 e y sin x2 y 3 3y 2 1 y Linearization L x y f 1 1 fx 1 1 x 1 fy 1 1 y 1 1 2 x 1 y 1 2x y fx 8 10 points extra credit points only if you get 90 or more points on problems 1 7 The difference r between the positions of two bodies moving in their gravitational field can be rescaled to satisfy r r00 3 r Show that r0 r is a constant r0 r 0 r00 r r0 r0 r00 r r 3 r r 0
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