MATH 115 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  Find  for the following

A)                    1 - x²
      y  =                       
                 1 + x²  

Solution

B)         y  =  (2 - 3x)¹⁴  

Solution

C)        y  =  x² sin(x - x⁴ )  

Solution

D)       3x² - x²y + y⁴  =  -3   

Solution

PROBLEM 2

Find the second derivative of the function below

Solution

PROBLEM 3 

A zoo keeper plans to build a rectangular cage for her newly arrived panda bears.  The front of the cage is to be made of glass at a cost of $100 per foot and the other three sides of the cage are to be made of concrete costing $50 per foot.  If the zoo keeper has $3000 to spend on the cage, what dimensions should the cage be so as to maximize the area of the cage?

Solution 

PROBLEM 4

The nearby sewage plant has just announced that it had a chlorine gas spill.  The gas cloud is in the shape of a hemisphere with equator on the ground.  The volume of the cloud is expanding at a rate of 10,000 cubic feet per minute.  How fast is the radius of the cloud moving by the time the cloud has reached the campus gym which is 200 feet away?  (Hint:  the volume of a hemisphere is 

                        2p r³ 
           V  =                      
                           3

and you want to find out how fast the radius is changing.)

Solution

PROBLEM 5

Without the use of the graphing capability of a graphing calculator, use calculus to find where the following function is increasing and decreasing.  Also use the first derivative test to locate and classify the relative extrema. 

        y  =  2x³ - 3x²

Solution

PROBLEM 6

The profit made is selling x automobiles can be modeled by the equation

        P  =  x²(8x² + x) ^-1/2

Solution

Find the marginal profit when x = 1.

PROBLEM 7

Use calculus to find the absolute extrema of 

        y  =  x³ - 12x

on the closed interval [0,3].