12.2 Calculus of Vector Functions by SGLee - HSKim - JHLee
1-5. Find 
and draw the position vector 
and the tangent vector 
for the given value of 
.
1. 
http://matrix.skku.ac.kr/cal-lab/cal-13-2-1.html
Calculus-Sec-12-2-Solution
12.2 Calculus of Vector Functions by SGLee - HSKim - JHLee
1-5. Find and draw the position vector and the tangent vector for the given value of .
1.
http://matrix.skku.ac.kr/cal-lab/cal-13-2-1.html
So, we can see that 
.
So, we can see that 
So, we can see that 
.
So, we can see that 
.
So, we can see that 
.
6-10. Find the derivatives of the vector functions.
6. 
.
7. 
.
8. 
.
9. 
.
10. 
.
11. 
, 
,
,
Hence 
. <-- r '(0)/abs(r'(0))
12. 
, 
.
.
Therefore, 
.
13. 
, 
,
,
.
14. If 
, find 
, 
, 
, and 
.
, 
.
Therefore, 
,
.
.
15. If 
, find 
and 
.
.
.
.
.
.
.
16-19. Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point.
16. 
; 
<-- z=t^4
We can see that 
.
For 
, 
. So, 
.
Hence 
.
17. 
; 
We can see that 
.
For 
, 
.
Therefore, 
.
Hence 
. <-- <1, 1, 0> + s <-1, -1, 1> = < 1-s, 1-s, s>
18. 
, 
, 
; 
We can see that 
.
For 
, 
. So, 
.
Hence 
.
19. 
, 
, 
; 
We can see that 
.
For 
, 
. So, 
.
Hence 
.
20-21. Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point.
Illustrate by graphing both the curve and the tangent line together.
So, parametric equation is 
.
So, parametric equation is 
.
22-24. Determine whether the curve are smooth.
22. 
. Since 
, this curve is not smooth.
23. 
.
First coordinate of 
cannot be zero for any real number 
, so this curve is smooth.
24. 
.
Second coordinate of 
cannot be zero for any real number 
, so this curve is smooth.
25-30. Evaluate the integrals.
25. 
.
26. 
.
27. 
.
28. 
where 
is a constant vector.
29. 
where 
is a constant vector. <-- e^(2t) j
30. 
.
31. Find 
if 
and 
.
, where 
is a constant vector.
Since 
, we finally have 
.
32. Find 
if 
and 
.
, where 
is a constant vector.
Since 
, 
.
So, we finally have 
.
33. If 
and 
,
use Theorem 4(Rules of differentiation) to find 
, 
.
, 
.
So, 
.
34. If 
and 
,
use Theorem 2(Rules of differentiation) to find 
.
,
,
,
.
35. Show that if 
is a vector function such that 
exists, then 
.
Proof
Using Theorem 4(Rules of differentiation),
.
36. If 
, show that 
.
Proof
We already know that 
.
Differentiate both sides, 
.
So, we finally have 
.
37. Find unit tangent vector to the curve 
, 
, 
at any time 
.
http://matrix.skku.ac.kr/cal-lab/cal-14-1-19.html
And we can draw a curve with given unit tangent vector.
, 
, 
, 
, 
, 
, 
; 
.
, 
, 
; 
.