MATH 106 PRACTICE FINAL

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 

Evaluate the given integrals, derivatives and limits.

A)   (20 Points)   

  Solution

B)    (20 Points)   

  Solution

C)   (20 Points)       

  Solution

D)   (20 Points)  

  Solution

E)    (20 Points)   

  Solution

F)    (20 Points)   

  Solution

G)   (20 Points) 

  Solution

H)   (20 Points)     

  Solution

PROBLEM 2

Set up the integrals that solve the following problems.  Sketch the appropriate diagram for each. Use a calculator to evaluate the integral.

A)   (15 Points)  Find the volume of solid that is formed by revolving the region bounded by y  =  x² - 4x and y  =  2x - 5 around the y-axis.

Solution

B)    (15Points)  Find the volume of the solid that is formed by revolving the region bounded by y  =  x² + 1 and y  =  5  around the line y  =  10 .

Solution

PROBLEM 3 (30 Points)

A force of 80 Newtons stretches a spring 70 centimeters on a mechanical device for driving fence posts.  Find the work done in stretching the spring the required 70 centimeters.

Solution

PROBLEM 4  (11 Points Each)

In 1960 the world population reached 3 billion people and in 1999 the population reached 6 billion people.

1. Write down the differential equation that reflects that the rate of population growth is proportional to the population.
   Solution

2. Solve this differential equation and use your solution to predict the population in the year 2050.
   Solution

3. It has been said, “It's the top of the ninth and humanity has been hitting nature hard. But we must always remember that nature bats last."  In particular environmentalists have warned that the carrying capacity of the earth is 10 billion people.  With this in mind it is better to use the model that the growth in population is proportional to the product of the population and 10 billion minus the population.  Write down a differential equation that reflects this statement.
   Solution

4. Solve this differential equation and use your solution to predict the population in the year 2050.
   Solution

 PROBLEM 5

Consider the integral

         
Use the trapezoidal rule with n = 4 to approximate this integral and sketch the indicated area and trapezoids.  

     Solution

PROBLEM 6 

Please answer the following true or false.  If true, provide an explanation.  If false provide an explanation or counter-example.

A.     (15 Points)  If f(x) is a continuous positive function then the surface area generated by revolving the curve y  =  f(x) for 0 < x < 1  about the x-axis is equal to the surface area generated by revolving the curve y  =  f(x) + 1 for0 < x < 1 about the x-axis
Solution

B.     (15 Points)  An aquarium has two exhibits each with vertical windows of the same area such that the bottoms are at the same depth and the tops are at the same depth.  Then even though the shapes of the two windows may be different, the fluid forces exerted on the windows are equal.
Solution