Unformatted text preview: 240 4. VectorValued Functions
3.. (sin 3t, cos 3t,2t3f2),0 g t g 1 4.(i+1,[email protected]+7,§t2);1gig 2; 5. (anti). 1 s t 52, 6. (t,tsint,tcost);0 g t 5 11' deﬁned by c(t) = {2 cost,2sint.t..
,if2a' 3: g 4n. t): (1,3t2,t3),if0§tg 1 a 7. Find the length of the path eft),
D g t 5 2a and C(t) == (2,t — 211,16) 8. Find the length of the path C(t), where cf
C(t) = team], if 1 g t g 2. _ ath C(t), deﬁned by s(t s the distance a particle traversing the trajectory 5 out at time a; that is, it gives ' d the arc length functions for  = (cos t, sint,t), with a : f}. I; UleNldT. mpresent
c will have traveled by time t if it start length of c between C(91) and C(t). Fin
curves aft] = [cosht, sinht,t) and ﬁt) 10. Let c be the path C(t) = (2t,t2, logt), deﬁned for t > 0. Find the
length of c between the points (2, 1,0} and [4,4, log2). 11. Find the arc length of the path C(t) : (t,tsint,tcos t) between (0.0. and (17,0, —1r]. 12. Let C(t) be a given path, a S t g b. Lets = aft) be a new variable, whe
is a strictly increasing C:l function given on [cu b]. For each s in [0501), of
there is a unique is with Mt) = 3. Define the function (1 : [15):(tt)I cc[b)1 —~
by d(s] = C(t). The path (1 is said to be a reparametﬁzation of c. (a) Argue that the image curves of c and d are the same. that c and d have the same arc length. (b) Show
(c) Let s = aft} = I: Hc’[r)lldr. Define d as above by d[s) : C(t). Sh
that d ‘ d is called the arc length reparametﬁzation of c. —» R3 be a smooth path. Assume c"(t) 75 0 for any t. 15. Let c : [ ,b]
(1 since RTE)“ = 1 vector (2’ (t) / Hc’ (t) 11 = T(t) is tangent to c at C(t), and,
is called the unit tangent to c.
= 0. [Hint Differentiate TUE) T(t) = 1.] (a) Show that T'(t}  T(t)
for T’ (t). (b) Write down a formula in terms of c ...
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 Summer '10
 CarolThomas
 Math

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