Calculus II - Review Worksheet  --  If you find any mistakes in the solutions, email me at abigailh@elmhurst.edu
  1. Sketch the following functions and state the domain and range of each.

    2.  

     

    a.                                                                              b. 

                        
     

        Domain is (-¥ , ¥ )                                                                       Domain is (-¥, ¥ )
        Range is [2, ¥ )                                                                             Range is (-¥, ¥ )
     
     
     
     

    c. 

                Domain is (-¥ , ¥ )
                Range is [- 1, 1]
     
     

  2. Find the line tangent to each function at the given x-value.
    1. at x = 2
         

         
        So, the slope of the tangent line is - 12.
        Now, a point on the line is (2, - (2)3) = (2,- 8).
        So, the equation of the line is , which simplifies to .
         
    1. at x = p / 4
         
        So, the slope of the tangent line is .
        Now, a point on the line is (p / 4, cos(p / 4)) = (p / 4,).
        So, the equation of the line is , which simplifies to .
         
    1. at x = 1

So, the slope of the tangent line is 0. This means that the tangent line is horizontal. Since it goes through the point (1, 81), the equation of the tangent line is y = 81.
 
  1. Evaluate the following integrals.

    2.  

     

    a.                                                                              b. 

                                                         
     

    c.                                                              d. 
     

                                                                       
                
     

    e.                                                              f. 

                              
     
     

  2. Express the area of the shaded region on the right as:
  1. a  limit of a sum.

    2.  

     

                    
     
     

  2. a definite integral.
          
  1. Consider the following polynomials:
, , 
  1. What should be?

    2.  

         
  2. Graph through all together on the calculator using a window that goes from x= -6 to x = 6 and from y = - 2 to y = 2. Can you see what function these polynomials are tending towards? If not, erase all of the graphs except . What function do you see? Graph this function along with to see how close they are. Now zoom out to see that they are not so close after all.

    3.  
        The graph of should be very close to the graph of sin(x) in the window given.
         
  3. How many terms do you think we would need to get an exact match?
You would need an infinite number of terms to get the function sin(x).
 
 
 
  1. a. If , find 
  1. If , find .

    2.  

           
           
  2. If , find