Calculus 3  Hour Exam 1   -   Math 241     Spring, 2010

 

 

1.

 

 

Find an equation of the plane passing through the point (2, -2, 1), and the line
x = 2 + 2t, y = 1 – t, z = 1 + t.

                                                 Answer:  x – 2z  =  0

 

 

2.

 

Sketch the surface   x2 –  y2  + 2y  +  4z2  = 0          Answer:  Hyperboloid of two sheets

 

 

 

 

 

 

3.

 

 

True or False?  Justify your answers.

 

a.        If  a x b  =  a x c    and   a  ≠ 0,  then  b  =  c           Answer:  False

 

b.      Let  v(t)  and  a(t)  be the velocity and acceleration of a moving particle at time

t.  If  a(t) is perpendicular to v(t) for all t, then the speed | v |  is constant.

 

Answer:  True

 

 

 

 

4.

 

 

Find the unit tangent vector, T, the principal normal unit vector, N, and the curvature, κ ,

for the plane curve  x= sin t, y = 2 cos t at the point where  t  =  π / 4.  Is the curve

concave up or down at that point?

 

Answers:  T = < 1, -2 >/√5   N =  < -2, -1 >/√5      κ  =  8 / (5 √10)     Concave down

 

 

 

 

 

5.

 

 

Let  v  =  < 1, -1, -1 >  and  a  =  < 5, -6, -1 >  be the instantaneous velocity and

acceleration vectors of a moving particle at time t.  Find the unit tangent and principal

unit normal vectors,  T  and  N,  the tangential and normal components of acceleration, 

aT   and  aN,  and the curvature  κ  at that moment of time t.

 

Answers:   T =  < 1, -1, -17 > / √3,  N =  < 1, -2, 3 >/ √14,  aT = 12/√3,  aN = √14

 

                   κ  =  √14 / 3

 

 

 

 

6.

 

A projectile is fired from the ground with an initial speed  vo and angle of elevation  α .

Find the distance,  X,  from the point the projectile is fired to the point it hit the ground  

in terms of  vo,  α,  and  g.  Assume that the motion is affected by gravity only.  You are

not allowed to use specific formulas from textbooks.

 

Answer:    X  =  2 vo2  sin α cos α / g