Math Club -- Test Bank
Professor R. Evans
(Fall, 97)

Math 10B Midterm #2, November 21,1997.  Direction :  Show Work!!


(1)  Find all inflection points of the function y = 4 x^6 - x^4.
     Give exact answers, and show in detil that your answers are
     genuine inflection points.    (20 pts)


(2)  At time t=0, water begins pouring at a constant rate into a hole
     carved on top of a hollow pumpkin.  The pumpkin's shape is roughly 
     spherical with radius 10 inches.  The pumpkin fills up at time t=8 
     seconds.  Sketch a rough graph of the depth of water in the pumpkin
     at time t, and label (with both coordinates) the inflection point 
     on your graph.  (10 pts)


(3)  Let y = x^3 - x.  Give the exact interval [a,b] on the x-axis
     where the function y is DECREASING.  (Be sure to show how you
     found your two numbers a and b.)  (10 pts)


(4)  Let y = x^3 e^(cx), where c denotes some fixed nonzero real number.
     (A)  Discuss the behavior of y as x tends to minus infinity. (10 pts)
     (B)  Assume now that the fixed number c above is NEGATIVE.  Find both 
          coordinates of the critical point not equat (0,0) for the 
          function y, and decide if it is a local min or a local max.  You
          may use either the first or second derivative test to decide.  
          Give detailed explanations.  Note:  The letter c should appear 
          in your answer for the critical point.  (20 pts)


(5)  The demand function for a certain product is q = 1700 - p^2 items, 
     where p is the dollar price per item.
     (A)  At what price p is revenue maximized?  (10 pts)
     (B)  Is the demand at price p = $25 elastic or inelastic? Explain 
          briefly.    (10 pts)


(6)  Suppose that f(x) is some given polynomial function of x.  Explain in
     WORDS (not in mathematical symbols) the procedure you would use on 
     this function f(x) to find its GLOBAL maximum on the interval [1, 5].
     Be brief but precise.  (10 pts)