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MATH 107 PRACTICE MIDTERM II

Please work out each of the given problems.  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.     (15 Points)  If A, B, and C are points, v is the vector from A to B, w is the vector from B to C, and v x w  =  0 , then A, B and C are collinear.
Solution

B.      (15 Points) If x  =  x(t),  y  =  y(t) are parametric equations of a line then dx/dt  is a constant.  
Solution

PROBLEM 2 (21 Points)

Consider the surface x² + z² - e^2y  =  0 .  This surface is formed by revolving a generating curve about an axis.  Find an equation of this generating curve and state the axis of revolution.
Solution

PROBLEM 3  (21 Points) 

Use vectors to find the equation of the line that passes through the point (2,3,4) and is perpendicular to the plane 5x - 4y + 2z = 7.

Solution

PROBLEM 4  (21 Points) 

Find all points (if any) of horizontal and vertical tangency.  Make sure to present your answer by listing the points not just the values of q.

        x  =  cos q        y  =  2sin(2q)

 Solution

PROBLEM 5 (21 Points) 

Determine the area of the first quadrant loop of  r  =  3sin(2q)

Solution

PROBLEM 6  (21 Points) 

Show that the polar equation for the hyperbola

           x²          y²
                   -            =  1          
           a²          b²  

   is

                        -b²
            r²  =                      
                      1 - e² cos² q 

given that 

                              b²
            e²  =  1 +            
                               a² 

  Solution

PROBLEM 7  (21 Points) 

Use vectors to determine if the triangle with vertices (1,0,1), (2,1,0), (0,0,4) is a right triangle.

 Solution

PROBLEM 8  (21 Points) 

Find parametric equations for the a particle moves along the line through (1,4,2) and (3,5,7) such that it is at the point (1,4,2) when t = 0 is at the point (3,5,7) when t = 2 and is speeding up as time progresses  

Solution