Calculus 3

MTH 230 Project Problem 3 Name: _________________________

1. Given  2

2),2sin(8),2cos(8)( ttttr , find A) the unit tangent vector B) The arclength on the

interval [0,t]. C) The arclength on the interval [0, 2]. (Show all definitions and work.)

2. A) Use the formula 3

r

rr



  to find the curvature at t =

2

 and at t =2. B) Find the radii of the

osculating circles at t = 2

 and at t =2. (Show all definitions and work.)

3. Use Mathematica to find the Unit Normal Vector for  2

2),2sin(8),2cos(8)( ttttr .

4. Use the results in problems 1. and 3. to find the Binormal Vector at t = 2

 . (Show the definitions and

work.)

5. Find the equation of the Normal Plane at t = 2

 .(Show the definitions and work.)

6. Find the equation of the Osculating Plane at t = 2

 . (Show the definitions and work.)

7. Given r(t) = ttt

etete ,cos,sin find A) the unit tangent vector B) The arclength on the interval [0,t].

C) The arclength on the interval [0, ln4]. (Show all definitions and work.) 8. A) Reparameterize the curve in problem 7. according to arclength. B) Find the location when one has traveled a distance of ln2 from 0 on this curve.

9. Given r(t) = ttt

etete ,cos,sin the Unit Normal Vector. (Show the definitions and work. This

requires some patience, but does work out.)

10. Find the curvature at t = ln4 on the curve r(t) = ttt

etete ,cos,sin .

11. Given a cycloid x = a(t – sint) and y = a( 1 – cost) which the path traced by a point on a rolling circle.

The interval [0, 2] produces one arch of the cycloid. Show the work that proves that the arclength of one arch is 8a.

12. State the domain and ranges of  2

2),2sin(8),2cos(8)( ttttr .

13. State the domain and ranges of r(t) = 2

2 22

7 2 , ,

8 88

t t

t tt   .

14. Find the 2

2 22

7 2 lim , ,

8 88 t

t t

t tt 

  .

15. Find the velocity, acceleration, and speed of a particle with the given position function.

2 ( ) 4 ,sin(2 ) 2 cos( ), cos(2 ) 2 sin( )r t t t t t t t t  

16. Read the parametric equations for trajectory motion on page 889. A projectile is fired with an initial speed of 180 m/sec and angle of elevation 54º. A) Find the range of the projectile B) the maximum height reached and c) the speed at input. 17. Read page 890 and find the tangential and normal components of the acceleration vector.

2 ( ) , ,3r t t t t