MATHEMATICS 317 Solutions to Ass1 - MATHEMATICS 317 Solutions to Ass#1 1(a r(t = x(t)i y(t j r(t = x(t)i y(t j = xi 2 x 2 j x = x(1 y =-2 x 2 2 1-2t e b

# MATHEMATICS 317 Solutions to Ass1 - MATHEMATICS 317...

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MATHEMATICS 317 Solutions to Ass. #1 1(a) . ) ( Hence . 0 , 1 ) 0 ( ; , constants ed undetermin for 1 ) ( i.e., , 1 ] (2) Eqn. [from (1) Eqn. ) 2 ( / 2 ) 1 ( / 2 ) ( ) ( ) ( ; ) ( ) ( ) ( 2 2 2 2 2 2 2 j i r j i r j i r j i j i r j i r t t t t t t e e t b a b a b e a ae t b e a y ae x x y x x x x t y t x t t y t x t - - - + = = = + = + + = + = = - = = - = + = + = & & & & & (b) . 4 ) ( ) ( ) ( 2 ) ( ) ( j i r v a j i r v t t t t e e t t t e e t t - - + = = = - = = & & & & 2. (a) , .... 1 , 0 , 2 0 sin , 1 cos 0 ) ( ; sin ) cos 1 ( ) ( ± = = = = = - = n n t t t t t t t π v j i v (b) The speed of the particle is given by . cos 2 2 ) ( t t - = v The maximum speed occurs when ,... 1 , 0 , ) 1 2 ( ± = = n n t π 3. . / ) / ( ) / / ( / 2 2 2 2 dt d dt d dt d dt d dt d a a a a a a b × = × + × = 4. . 3 3 3 3 2 2 2 2 2 2 2 2 dt d dt d dt d dt d dt d dt d dt d dt d dt d dt d dt d dt d r r r r r r r r r r r r r r r × = × + × + × = × 5. (a) From Property 5: . )] ( [ ] ) [( ) ( ) ( d b a c d c b a d c b a × × - = × × = × × Then using Property 6, one has ). )( ( ) )( ( ] ) ( ) [( )] ( [ d a c b d b c a d b a c a b c d b a c - = - - = × × - (b) A = area of triangle = ). ( ) ( 4 1 2 2 1 2 1 2 1 2 1 r r r r r r × × = × A