Vectors Calaulus 1.15

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.

  

PROBLEM 1 

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

 

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

Solution

True,  since 

        s(t)  =  t

We have 

        s''(t)  =  0

So that 

        a  =  s''(t)T + k(s')²N  =  k(s')²N  

so that a and N are multiples of each other.  Hence they are parallel.

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

       

Solution

False,  for example if

        r(t)  =   i + t j

then

       

and

        ||r'(t)||  =  || j ||  =  1

So they are different. 

PROBLEM 2 

 

Let

        r(t)  =  2t i - 4t² j

Find T(-1)

Solution

We have

        r'(t)  =  2 i - 8t j 

so that 

       

Now just plug in -1 for t to get

          

Find N(-1).
   

Solution

 

Use the quotient rule to get

       

Now get rid of the denominator and multiply by the root to get

        (-4j)(1 + 16t²) - (i - 4tj)(16t)

        =  4[-j(1 + 16t²) - (i - 4tj)(4t)]

Now divide by 4 and plug in -1 for t to get

        -17 j - (i + 4j)(-4)  =  4i - j

Dividing by the magnitude gives

       

 

Find the equation of the circle of curvature for r(t) at t  =  -1.  

The acceleration vector is 

        a  =  r''(t)  =  -8j 

Dotting with the normal vector gives the component of the acceleration in the direction of the normal vector.  We get

        a_N  = 

and

        (ds/dt)²  =  ||r'(t)||²  =  1 + 64t² 

evaluating at t  =  -1 gives

        65

Using the curvature formula gives

        65K  =  a_n  

so that 

       

The radius of the circle is just the reciprocal of the curvature.

To find the center, we add the vectors

        Center  =  r(-1) + KN

which gives

       

Call this 

        ai + bj 

Then the equation of the circle is just

        (x - a)² + (y - b)²  =  1/K²