Maths Problem Set 1

Prove that  is one-to-one function.                 

Let f:, as given below.  Is f a one-to-one function?  Please explain why or why not.                                                                                                                                 

The modulo function (a mod n or a modulo n) maps every positive integer number to the remainder of the division of a/n.    For example, the expression 22 mod 5 would evaluate to 2 since 22 divided by 5 is 4 with a remainder of 2.  The expression 10 mod 5 would resolve to 0 since 10 is divisible by 5 and there is not a remainder.

Find    

Find                 

Find           

Find   as       

Prove that  is a continuous function. 

Find derivatives of the following functions using differentiation rules.  (Do not use the definition of a derivative!)

Find the derivative of the following function at :     

Find the derivative of the following function   using the definition of a derivative.  (Hint: though you can use the rules of differentiation to check your answer, you must use the definition of a derivative to solve this problem in order to receive any credit for your response)   

Let   and , where.   Find  and as well as the domain and range of these functions.

Find the derivative of  by using the definition of the derivative