hwsol_11

# hwsol_11 - Homework Assignment 11(11.5 Solutions Page 902 6...

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Homework Assignment 11 - (11.5) - Solutions Page 902: 6*, 10*, 14*, 16*, 18*, 20*, 22*, 26*, 29-32, 34*, 41(extra points), 42(extra points) *Let r(t) ! (t,sin(t),2t). Find the radius and center of the osculating circle at t ! pi/2. 6. r " ! t " !# t , t 3 \$ , t ! 0, t ! 1 r " % ! t " !# 1,3 t 2 \$ , r " % ! t " ! 1 & 9 t 4 , T " ! t " ! 1 1 & 9 t 4 # t 2 \$ T " ! 0 " !# 1,0 \$ , T " ! 1 " ! 1 10 # \$ T " % ! t " ! ! 36 t 3 2 ! 1 & 9 t 4 " 3/2 # t 2 \$& 1 1 & 9 t 4 # 0, 6 t \$ ! 1 ! 1 & 9 t 4 " 3/2 # ! 18 t 3 , ! 54 t 5 & 6 t ! 1 & 9 t 4 " \$! 1 ! 1 & 9 t 4 " 3/2 # ! 18 t 3 ,6 t \$ T " % ! 0 " !# 0,0 \$ , N " ! 0 " !# \$ T " % ! 1 " ! 1 10 # ! 18, 6 \$ , T " % ! 1 " ! 1 10 18 2 & 6 2 ! 6 N " ! 1 " ! 1 6 1 10 # ! 18, 6 \$! 1 10 # ! 3, 1 \$ 10. r " ! t " !# cos t , sin t , sin t \$ , t ! 0, t ! ! 2 r " % ! t " !# ! sin t , cos t , cos t \$ , r " % ! t " ! sin 2 t & cos 2 t & cos 2 ! 1 & cos 2 t T " ! t " ! 1 1 & cos 2 t # cos t , sin t , sin t \$ T " ! 0 " ! 1 2 # 1,0,0 \$ , T " ! 2 !# 0,1,1, \$ T " % ! t " ! ! ! 2cos t sin t 2 ! 1 & cos 2 t " 3/2 # cos t , sin t , sin t 1 1 & cos 2 t # ! sin t , cos t , cos t \$ ! 1 ! 1 & cos 2 t " 3/2 cos t sin t # cos t , sin t , sin t ! 1 & cos 2 " # ! sin t , cos t , cos t \$ ! 1 ! 1 & cos 2 t " 3/2 # cos 2 t sin t ! sin t ! cos 2 t sin t , cos t sin 2 t & cos t & cos 3 t , cos t sin 2 t & cos t & cos 3 t \$ ! 1 ! 1 & cos 2 t " 3/2 # ! sin t , 2cos t t \$ T " % ! 0 " ! 1 22 # 0, 2,2 \$! 1 2 # 0,1,1 \$ , T " % ! 0 " ! 1 2 2 ! 1, N " ! 0 " ! 1 2 # \$ T " % ! 2 !# ! \$ , T " % ! 2 ! 1, N " ! 2 !# ! \$ 14. r " ! t " !# t , t 3 \$ , t ! 0 From Problem 6., we have the following r " % ! t " !# t 2 \$ , r " % ! t " ! 1 & 9 t 4 , r " % ! 0 " ! 1 T " % ! 0 " !# \$ , N " ! 0 " is not defined. 1

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Since T " % ! 0 " ! 0, " ! 0. So there is no osculating circle at t ! 0. 16. r " ! t " !# 2cos t , 3sin t \$ , t ! ! 4 r " % ! t " !# ! 2sin t , 3cos t \$ , r " %% ! t " !# ! t , ! 3sin t \$ , r " % ! t " ! 4sin 2 t & 9cos 2 t r " % ! t " r " %% ! t " ! i " j " k " ! t 3cos t 0 ! t ! t 0 ! 6 k " , r " % ! t " r " %% !
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hwsol_11 - Homework Assignment 11(11.5 Solutions Page 902 6...

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