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MATH 115 MIDTERM III

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 Without the use of your graphing calculator,

A)   Determine the relative extrema if any.

B)    Determine where the function is increasing and decreasing.

C)   Determine the inflection points if any.

D)   Determine where the function is concave up and concave down.  

E)  Use parts A through D to sketch the graph of the function.

I.               y  =  x³ - 12x

Solution

II.            y  =  3x⁴ - 4x³

Solution

III.                      3x - 4
       y  =                               
                     4x - 3

Solution

IV.                       x
       y  =                              
                    x² + 1  

Solution

PROBLEM 2  You manage the new Spockwood ski resort.  Your research shows that if you charge $40 per lift ticket, you can expect to sell 3000 tickets.  For every $5 increase in price you can expect to lose 250 customers.  Assume the demand equation is linear.

A)    Find the demand equation.  

Solution

B)     What price will yield the maximum revenue?  

Solution

C)     Is the demand elastic or inelastic when the price is $45?

  Solution

PROBLEM 3   At the blood bank, a cylindrical bottle without top is to contain 100 cc of blood.  What should the dimensions of the bottle be in order to minimize material costs?  Assume that the glass has the same thickness throughout.

Solution

PROBLEM 4  The department of fish and game have introduced a species of trout that was once native to the region.  The number (in hundreds) of this species t years after their introduction can be modeled by the equation

                        90(2 + 5t)
         P(t)  =                                 
                           1 + 10t
Use differentials to approximate the change in the trout population from three to 3.5 years after their introduction.

  Solution