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.

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.

20.

We can see that .

So, parametric equation is .

21.

We can see that .

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

Differentiate both sides, .

So, we finally have .

37. Find unit tangent vector to the curve  at any time .

And we can draw a curve with given unit tangent vector.