# MATH 2D F14: Midterm 1

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Midterm 1 Solutions for MATH 2DHamid Hezari

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. 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 (

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2 8 \

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 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 yv e l o c i t y v e c t o r u (

) 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 (

) 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

• ( (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 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) .

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u 9e j , 2 4 C 1 c )-

A /

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- 2 [ t h z t\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 .

7 \ l ( f v in w + A

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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