Thomas Calculus 13th [Solutions]

934 Chapter 13 Vector-Valued Functions and Motion in Space
1 1 1
6. 2 3 1 1
3
i k dt 2sin t i 3 tan t k i k
0 2 2
1 t 1 t
0 0
4
7.
8.
1 2 2
t t 1 1
1 t t 1
te e dt e e t e 1 e 1
0 2 0 0 2 e
0
i j k i j k i i k
ln 3 t t t t
ln 3
t
ln 3
ln 3
te e ln t dt te e e t ln t t
1 1 1
1
i j k i j k
3 ln 3 1 i 3 e j ln 3 ln(ln 3) 1 1 k
9.
10.
/2 2 /2
1 1
0 cos t i sin 2 t j sin t k dt
0
cos t i sin 2 t j
2 2
cos 2 t k dt
/2 1
/2
1 1
/2
sin t cos t t sin 2t
0 2 0 2 4 0
4
i j k i j k
/4 2 /4
2
sect i tan t j t sin t k dt sect i sec t 1 j t sin t k dt
0 0
/4 /4 /4
ln sect tan t tan t t t cost sin t ln 1+ 2 1
1
0 4 4 2 2
i j k i j k
11.
2 2 2
r ti t j tk dt t
i
t
j
t
k C ; (0) 0 0 0 2 3 2 3
2 2 2
2 2 2
t 1 t 2 t 3
2 2 2
r i j k
r i j k C i j k C i j k
12.
2 2 2 16 3
180t 180t 16t dt 90t 90 t t ;
3
r i j i j C
2 2 16 3
100 90t 90t t 100
3
C j r i j
2 2 16 3
(0) 90(0) 90(0) (0) 100
3
r i j C j
13.
3 1/2 t 1
3/2 t
( t 1) e dt ( t 1) e ln( t 1) ;
2 t 1
r i j k i j k C
3/2 0 3/2
t
(0) (0 1) e ln(0 1) ( t 1) 1 1 e 1 ln( t 1)
r i j k C k C i j k r i j k
14.
4 2 3
3 2 2
t 4 t ) t 2t dt t 2 t t 2t
;
4 2 3
r i j k i j k C
4 2 3
t 2
2t
1 t 1 2t
4 2 3
C i j r i j k
4
2
2 2(0)
(0) 0 2(0) 0
3
4 2 3
r i j k C i j
dr
dr
dr
15. ( 32 k) dt 32 tk C1;
(0) 8i 8j 32(0) k C
dt
1 8i 8j C1
8i 8j 8i 8j 32 tk;
dt
dt
2
2
r 8i 8j 32tk dt 8ti 8t j 16 t k C2;
r(0) 100k 8(0) i 8(0) j 16(0) k C2
100k
2
2 100 8t 8t 100 16t
C k r i j k
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