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...

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 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 (6 points)


5.

Find , as (6 points)


6.

Find , as (6 points)






7.

Find , as (6 points)


8.

Prove that is a continuous function (8 points)


9.

Find derivatives of the following functions using differentiation rules


1.


(3 points)



2.


(3 points)



3.


(3 points)



4.


(3 points)



5. (3 points)




6. (3points)



7. (3 points)


10. Find the derivative of the following function at (5 points)






11. Find differentials of the following functions:



1. (4 points)



2. (4 points)



3. (4 points)


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





Homework for Lecture 1 Assignment 1 1. Let Z Z f ® : , where Z is the set of integers and 5 ()101 fxx =+ Is ) ( x f a one-to-one function? (10 points) 2. Prove that ()43 fxx =- 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, é ù x x f ® : , where é ù 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) 4. Find 3 2 lim 1 nn n + - when ¥ ® n (6 points) 5. Find 73 72 317 lim 21 xx xx ++ ++ , as ¥ ® x (6 points) 6. Find 2008 2009 1 lim 1 x x + + , as ¥ ® x (6 points) 7. Find 111 , 111 , 111 111 , 111 lim 3 4 + + x x , as 0 x ® (6 points) 8. Prove that 2 ()4 fxx =- is a continuous function (8 points) 9. Find derivatives of the following functions using differentiation rules 1. ()95 fxx =+ (3 points) 2. ()3 x fxe = (3 points) 3. 3 ()34 fxx =+ (3 points) 4. 42 ()3431 fxxxx =++- (3 points) 5. ()(23) x fxxe =+ (3 points) 6. 3 ()(21) x fxxe - =- (3points) 7. 2232 ()(2)(44)44 fxxxxxxxxx =-=-+=-+ (3 points) 10. Find the derivative of the following function at 0 1 x =- (5 points) 2 ()31 fxxx =++ 11. Find differentials of the following functions: 1. 2 ()31 fxxx =++ (4 points) 2. 1 () x fxe - = (4 points) 3. 3 ()4 x fxex =-+- (4 points) Complementary problems 12. Let x x f = ) ( and 2 ) ( x x g = , where R x Î Find g f o and f g o as well as the domain and range of these functions. 13. Prove using the definition that 0 1 lim 2 = n , when ¥ ® n 14. Find the derivative of x x f 1 ) ( = by using the definition of the derivative ​© 2012 Trustees of Boston University. Materials contained within this course are subject to copyright protection. _1299680641.unknown _1299681524.unknown _1299681909.unknown _1299683753.unknown _1299683799.unknown _1299683920.unknown _1299683929.unknown _1299683772.unknown _1299682228.unknown _1299682620.unknown _1299682848.unknown _1299682609.unknown _1299682139.unknown _1299681730.unknown _1299681867.unknown _1299681618.unknown _1299681335.unknown _1299681405.unknown _1299680985.unknown _1162887111.unknown _1162887289.unknown _1165502300.unknown _1166693289.unknown _1166693449.unknown _1165471999.unknown _1162887113.unknown _1162887248.unknown _1162887112.unknown _1161410688.unknown _1161410690.unknown _1161410693.unknown _1161410909.unknown _1161410692.unknown _1161410689.unknown _1156268419.unknown _1156268467.unknown _1156268395.unknown