A. Find the arc length function for the curve measured from the point P in the direction of increasing t and then reparametrize the curve with respect to arc length starting from P. b. Find the point 4 units along the curve (in the direction of increasing t) from P.

r(t) = et sin t i + √2 et k, P(0, l ,√2)

The correct question has r(t) = e^t sint i + e^t cost j + √2 e^t k

(a) 2e^t

(b) r(t) = (L/2 + 1)sin(ln|L/2 + 1|) i + (L/2 + 1)cos(ln|L/2 + 1|) j + √2(L/2 + 1) k

Step-by-step explanation:

Given r(t) = e^t sint i + e^t cost j + √2 e^t k

(a) To find the arc length of the function, we first differentiate r(t) to obtain r'(t). Doing this, we have

r'(t) = (cost + sint)e^t i + (cost - sint)e^t j + √2 e^t k

Next, we find the magnitude of r'(t). Doing this, we obtain

|r'(t)| = √{[(cost + sint)e^t]² + [(cost - sint)e^t]²+ (√2 e^t)² k}

= √(4e^(2t))

= 2e^t

(b) Now, because t = 0 corresponds with the (0, 1, √2), we have the arc length function, L to be the integral from 0 to t of |r'(v)|dv

L = integral of 2e^v dv from v = 0 to t.

Evaluation the integral, we have

L = 2e^t - 2e^0

= 2e^t - 2

L = 2(e^t - 1)

Let us make t the subject of the formula.

Divide both sides by 2

L/2 = e^t - 1

Adding 1 to both sides

L/2 + 1 = e^t

e^t = L/2 + 1

Taking natural logarithm of both sides

t = ln|L/2 + 1|

Finally, we put t = ln|L/2 + 1| in the original equation, r(t) we were given to have

r(t) = (L/2 + 1)sin(ln|L/2 + 1|) i + (L/2 + 1)cos(ln|L/2 + 1|) j + √2(L/2 + 1) k

1. 3.4/4

2. 0.8/7

3. 0.5 x 3

4. 0.9 x 6

5. 10.8

6. 2.35

7. 0.567

8. 9.35

9. 0.02

10. 0.06

11. 0.16

12. 0.42

13. 0.63

14. 5.6

15. 8.1

16. 9.9

17. 0.28

18. 0.48

19. 0.27

20. 0.49

21. 0.45

22. 0.72

23. 0.9

24. 0.88

25. 0.75

26. 0.16

27. 0.328

28. 0.495

29. 0.72

30. 1.068

there are no multiple choices