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- . 9 \ Ou t an g ent line . Th e n elim i nate the p aram ete r to find the C artesian e q uation of this ( c ) F i nd the p aram etric e q uation of the tan g ent line at the p oint T ( ) 2 8 \ vector is alw a y s tan g ent to the cu]: ve th at y ou g oc he co rrect direction in p art ( a ) . ( N ote that the velocit y velocit y vector u ( ) o n y ou r g ra p h in p art ( a ) . U sc th i s t o con:irm ( b ) C o m p ute u ( t ) , t h e velocit y vector of ¡ . Pl ot the p oint T ( ) and its 1 . ( n ) L et c b e the cur ve defined b y

MATH 2D F14: Midterm 1

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Midterm 1 Solutions for MATH 2DHamid Hezari

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Page 1: MATH 2D F14: Midterm 1

-. 9 • \ Ou

t a n g e n t l i n e .

T h e n e l i m i n a t e t h e p a r a m e t e r t o f i n d t h e C a r t e s i a n e q u a t i o n o f t h i s

(c ) F i n d t h e p a r a m e t r i c e q u a t i o n o f t h e t a n g e n t l i n e a t t h e p o i n t T (î ) •

2 8 \ • ヒ

v e c t o r i s a l w a y s t a n g e n t t o t h e c u ] : v e

t h a t y o u g o c 七 h e c o r r e c t d i r e c t i o n i n p a r t (a ) . (N o t e t h a t t h e v e l o c i t y

v e l o c i t y v e c t o r u (î ) o n y o u r g r a p h i n p a r t (a ) . U s c t h is t o c o n : i r m

(b ) C o m p u t e u (t ) , t h e v e l o c i t y v e c t o r o f ¡ . P l o t t h e p o i n t T (î ) a n d i t s

1 . (n ) L e t c b e t h e c u r v e d e f i n e d b y

Page 2: MATH 2D F14: Midterm 1

( ¡ (c ) F i n d t i l e e q u a t i o n o f t h e p l a n e t h a t c o n t a i n s b o t h L r a n d L 2 •

0 (b F i n d t h e d is t a n c e b e t w e e n L 1 a n d L 2

Lt x ’ Lt A ’(

" A • - l 4 2

l i l l e L 2 t h a , t g o e s t l l m u g h p a n el i s p a r a ó l t o L 1 • B ;’0 (a) Sho w t h a t p i s i l o t o n L 1 . T h c n i n t l t h c

-p a r i l l 11M r i c c q u a t i m i o f t h c

2 . L c t p b e t h c p o i n t (1 , 0 ,- 1) 1 & . 1 l d L I b c t h e l i l i c (1 + t , 1 - t , 24) .

Page 3: MATH 2D F14: Midterm 1

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u 9e j , 2 4 C 1 c )

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\D (c ) C a l c u l a t e t h c a r c l e n g t h o f r (t ) i n t h e t i m e i n t e r v a l o [ t [ 5 .

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w s J + w < l L • u - 6 4 •

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(b ) Sk e t c h t h e c u ] : v e a n d t h e c y l i n d e r i n o n e g r a p h . U s e a r r o w s t o s h o w

e q u a t i o n o f t h a t c y l i n d e r .

5 (a ) Sh o w t h a t t h c Cl l l : v c c s i t s o n t h e s u r f a c e o f a c y l i n d e r a n d % n d t h c

r (t ) = s i n (24) , ; : , c o s (26)

3 . L e t c b o t , ll e ( •

1 1 > v : de f i n e d b y

Page 4: MATH 2D F14: Midterm 1