MATH 202 PRACTICE MIDTERM 1

 

Please work out five of the given six problems and indicate which problem you are omitting.  Credit will be based on the steps that you show towards the final answer.  Show your work.

    Printable Key

PROBLEM 1 Please answer the following true or false.  If false, explain why or provide a counter example.  If true explain why

 

A)   (12 Points)  If r(t) is parameterized by arclength, then a and N are parallel.

Solution

B)    (13 Points)  If r(t) is a differentiable vector valued function then

       

Solution

 

PROBLEM 2  (25 Points)

 

Let

        r(t)  =  2t i - 4t2 j

 

  1. Find T(-1).
         Solution
  2. Find N(-1).
         Solution
  3. Find the equation of the circle of curvature for r(t) at t  =  -1.  
    Solution

 

 

PROBLEM 3 (25 Points)  Jason Elam (the football kicker for the Denver Broncos) can kick a football with an initial velocity of 60 feet per second.  At what angle should the ball be kicked to maximize the horizontal distance that the ball travels before it lands on the ground?   (Use vectors please).

  Solution

 

PROBLEM 4 (25 Points) Prove the following theorem:

Let r(t) be a differentiable vector valued function, then

        |(r x v) . a|  =  ||r'|| ||aN|| |r . (T x N)|

  Solution

PROBLEM 5  (25 Points)

Find the parametric equations of the tangent line to the curve that is formed by intersecting the sphere  x2 + y2 + z2  =  2 and the plane x + y - z  =  2 at the point (1,1,0).

  Solution

PROBLEM 6   (25 Points)

If 

        a(t)  =  t i + j - k

find r(5) if 

        r(0)  =  i - k      and         r(1)  =  j + k

  Solution

Extra Credit:  Write down one thing that your instructor can do to make the class better and one thing that is going well.

(Any constructive remark will be worth full credit)