Mathematics 54 1st Semester A.Y. 2004-2005Last Long Examination 4 October 2004

This exam is for two hours only. Use only blue or black ink. Show your complete and organizedsolutions. Good luck!

1. Identify the domain of ~R(t) = ln t i+ sin1 (1 t) j + 11 t k. (2 points)

2. Analyze the continuity of

~R(t) =

et 1t

i+ t csc t j, if t 6= 0i+ j, if t = 0

at t = 0. (3 points)

3. If ~R(t) =1

1 + t2i+

t1 + t2

j and ~R(0) = j, find ~R(t). (4 points)

4. Express Dt[(

~R(t) ~Q(t)) ~S(t)

]in terms of ~R, ~Q, ~S and their derivatives. (3 points)

5. The position vector of a point P at any time t is given by

~R(t) =12cos

(t2)i 1

2sin

(t2)j +

13t3 k.

(a) Verify that the arc length s of ~R, where s is measured from the point where t = 0, is given by

s(t) =13(1 + t2

) 32 1

3.

(4 points)

(b) Find AT (0) and AN (0). (5 points)

6. Consider~R(t) =

( ta

cos ((u)) du)i+

( ta

sin ((u)) du)j,

where (u) is a differentiable function in some interval I containing a. Show that the curvature of ~Ris |(t)|. (5 points)

7. Consider~R(t) =

( t0

cos(u2)du

)i+

( t0

sin(u2)du

)j.

(a) Verify that the moving trihedral at t =pi

2consists of the vectors ~T =

22,

22, 0

,

~N =

22,

22, 0

and ~B = 0, 0 1. (7 points)

(b) Use the result in (6) to find the radius of curvature of ~R at t =pi

2. (2.5 points)

(c) Find the vector of curvature ~K at t =pi

2. (1.5 points)

8. The muzzle speed of a gun is 642 ft/sec. At what angle of elevation should the gun be fired so that

a projectile will hit an object on the same level as the gun at a distance of 256 feet from it? (6 points)

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9. Determine whether the given sequence is convergent or divergent.

(a){2n

n2

}+n=1

(2 points)

(b) {cothn}+n=1 (2 points)

(c){1 3 5 (2n 1)2 4 6 (2n)

}+n=1

(3 points)

K(t) =~R(t) ~R(t)

~R(t)3 ;~K(t) =

~T (t)

~R(t)

END OF EXAMTOTAL: 50 points

carmen/10032004

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