Assignment 11. Let , where is the set of integers and Is a one-to-one function? (10 points)2. Prove...

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Assignment 11. Let , where is the set of integers and  Is a one-to-one function? (10 points)2. Prove that is one-to-one function (10 points)3. The ceiling function maps every real number to the smallest integer greater than  or equal to that number, , where is the smallest integer greater  than or equal to . Is this function one-to-one? List five numbers that have  the same image ( 10 points)4. Find when       XXXXXXXXXX6 points)5. Find , as    XXXXXXXXXX6 points)6. Find , as        XXXXXXXXXX6 points) 7. Find , as  XXXXXXXXXX6 points)8. Prove that is a continuous function (8 points)9. Find derivatives of the following functions using differentiation rules1.                      XXXXXXXXXX3 points) 2.                        XXXXXXXXXX3 points) 3.                     XXXXXXXXXX3 points) 4.             XXXXXXXXXX3 points) 5.                   XXXXXXXXXX3 points)  6.                  XXXXXXXXXX3points) 7 XXXXXXXXXX3 points)10. Find the derivative of the following function at  (5 points) 11. Find differentials of the following functions: XXXXXXXXXXpoints) XXXXXXXXXXpoints) XXXXXXXXXXpoints)Complementary problems 12. Let and , where Find and as well as the domain and range of these functions.13. Prove using the definition that , when 14. Find the derivative of by using the definition of the derivative

Robert · Accepted Answer

© 2012 Trustees of Boston University. Materials contained within this course are subject to copyright 
protection. 
 
Assignment 1
1.  Let ZZf : , where Z  is the set of integers and 5( ) 101f x x   
     Is )(xf  a one-to-one function?                                             (10 points) 
 
 
The given function is an odd function . Any line drawn parallel to the y axis cuts the 
graph into one point only. Therefore the given function is a one to one function.
2.  Prove that ( ) 4 3f x x   is one-to-one function                   (10 points) 
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1 2 3
Series 1
Column2
Column1
 
© 2012 Trustees of Boston University. Materials contained within this course are subject to copyright 
protection. 
 
 
The graph of f(x) is a straight line .Any line drawn parallel to the y axis divides the 
graph into one point only. Hence the given function is a one to one function.
3.  The ceiling function maps every real number to the smallest integer greater than  
     or equal to that number,  xxf : , where  x  is the smallest integer greater  
     than or equal to x . Is this function one-to-one? List five numbers that have   
     the same image                                                                       ( 10 points) 
    yes the given function is one to one. 
Any line drawn parallel to the y axis divides the graph into one point only. 
Five numbers that have the same image are 1.2,1.3,1.4,1.5,1.6 
4.  Find 
3
2
lim
1
n n
n


 when n                                                 (6 points) 
Dividing numerator and denominator by n
2 
lim (n+1/n)/(1-1/n) 
     when n  
           =∞ 
5.  Find 
7 3
7 2
3 17
lim
2 1
x x
x x
 
 
,

Assignment 1 1. Let , where is the set of integers and Is a one-to-one function? (10 points) 2. Prove that is one-to-one function (10 points) 3. The ceiling function maps every real number to the...

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