Evaluate, if possible, the following limits. Use ∞ & – ∞ as necessary and show all work.

1.) 

	  2.) 

3.) 

	  4.)	

5.) 

	
  6.) 	

7.) Use the definition of continuity to prove if the function below is continuous at. 
  No graphical explanations!

8.)  a.) Sketch the function: 

    b.) Determine and prove where the function is continuous.

    c.) Determine and prove where the function is differentiable.

Evaluate, if possible, the following limits. Use ∞ & – ∞ as necessary and show all work.

9.)	

	10.)		
11.)	 (in terms of x)

	12.)		

13.)  Multiple Choice.  Show work for full credit.  
Which of the following is/are true about the function g if?

I.	g is continuous at
II.	The graph of g has a vertical asymptote at
III.	The graph of g has a horizontal asymptote at

a.)	I only	 b.) II only	       c.) III only	    d.) I and II only    	   e.) II and III only

14.)  Multiple Choice.  You must include work to justify your answer!!

At , the function given by  is
(A)	Undefined
(B)	Continuous but not differentiable
(C)	Differentiable but not continuous
(D)	Neither continuous nor differentiable
(E)	Both continuous and differentiable

15.) Sketch the graph of ONE function f (x) that satisfies all of the following conditions.

16.)  a.) Sketch the function: 

    b.) Determine and prove where the function is continuous.

    c.) Determine and prove where the function is differentiable.

17.) Prove the equation has only 1 real root.

18.) The functionis define for and has and. 
Use the Mean Value Theorem to find the smallest possible value for.

19.) Determine whether the series converges or diverges. If it converges, evaluate the integral (Exact value).

a.)				b.) 

20.) Determine whether the series converges or diverges.

a.)  				b.) 

21.) Evaluate the following convergent integral,  (Exact value).

22.)  a.)  Using summation notation (you do not have to evaluate), find L and U such that: 

    b.)  In terms of, evaluate the integral .

    c.)  Hence or otherwise, find the area under the curve  on the interval .

23.) Use the comparison test to determine whether the following integral converges or diverges.

24.) Find the upper and lower sums of the integral  (Summation notation only) 

Thus, find exact values for A and B such that .

Determine whether the following series converge or diverge. Show all work and clearly state which test(s) are used.

25.) 			

26.) 

27.) 

28.) 

Complete the following steps for the series below:

•	Find the first 4 partial sums.  
•	Determine whether the series converges or diverges. 
•	If the series converges, determine the sum of the series.

29.) 

For problems (2) – (7), determine whether the following series converge or diverge. Show all work and clearly state which test(s) are used.

30.)  	

31.)  

32.)  			    

33.)  

34.)  		    

35.)  

36.) Determine whether the series conditionally converges, absolutely converges, or diverges.

37.) Determine the interval of convergence for the series .

38.)  Find the Maclaurin series expansion of the function below up to theterm.

 

39.)  Find the Maclaurin series expansion of the function below up to theterm.

40.)  Find the first three terms of the Maclaurin series for .

41.)  Given that , 
a.	Show that the first two non-zero terms of the Maclaurin series for are .

b.	Use this series to find an approximation in terms of  for .

42.)  Find the Maclaurin series for  as far as the  term.

43.)  Use a Maclaurin series to approximate the integral, .  

44.)  Find the first 3 non-zero terms of the Maclaurin series for .

45.)  Find the Taylor polynomial of degree 2 for  centered about .

46.)  Find the Maclaurin series for up to and including the  term.   

47.)  Without using the any of the shortened formulas, find the Maclaurin series for the function  up to the  term.  Express your final answer as a series in the form .  
Note:  You must show the correct calculations including all necessary derivatives.

48.)  Find the first three nonzero terms in the Maclaurin series for .

49.) Find the 3rd degree Taylor series for the function  centered about .

50.) Use the first 4 terms of a Maclaurin series to approximate the integral, .  You may express your answer as a decimal or a fraction in reduced form.

Solve the following differential equations.
51.) 			     52.) 

53.) 		               54.)   (Explicit @ x=1,y=1)

55.) 			     56.) 

57.)			     58.)

51.) 				     52.)

51.) 				     52.)