Calculus 3  Hour Exam 1 (continued)  -   Math 241     Fall, 2010

 

 

3a.

 

 

If  a(t) = < 0,0,1 >  and   v(0) = < 1,0,0  >,   r(0) = < 0,1,0 >,   find  r(t).

 

                                                 Answer:   r(t)  =  <  t, 1, t2/2  >

 

 

 

3b.

 

Use   r(t)  to find  T(t) ,     the unit tangential vector at any time  t.

 

Answer:   T(t)  =  < 1, 0, t > / √ (1 + t2 )    

 

 

 

3c.

 

 

Find   aT  and  aN  at   t = 1.  (the tangential and normal components of acceleration)

 

Hint:  draw a picture.                   Answers:  aT  = 1/√2,  aN   = √(1/2)

 

 

 

 

 

 

4.

 

 

Joe the bug follows a trail of strawberry jam tracked across the kitchen floor by a four

year old along the path  x(t)  =  √(1 + t),  y(t)  =  4 + t2  .  The density of the jam at a

point is given by the function  JD(x,y).  Determine the change in jam density dJ/dt

 at  t = 8   given the following data.

∂JD/∂x(1,4) = 5             ∂JD/∂y(1,4) = 9                    

∂JD/∂x (3,68) = 6         ∂JD/∂y(3,68)  =  10

∂JD/∂x(2,3)  =  7           ∂JD/∂y(2,3)  =  11

JD (2,3) = 8           JD (1,4)  =  12                   Answer:  dJ/dt  =  161

 

 

5a.

 

 

Find the distance from (1,1,1)  to the plane  x + 2y + 3z  =  7.        Answer:  1/√14

 

5b.

 

If  r(t)  =  < 1, t, t2 > ,  find   ĸ (t). the curvature.       Answer:  ĸ (t)  =  2 / (1 + 4t2 )3/2