Maths Problem Set 1
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maths_problems_set1.docx
1. Prove that is one-to-one function.
2. Let f:, as given below. Is f a one-to-one function? Please explain why or why not.
( B ) ( A )
( f )
( 5 ) ( 5 ) ( 1 )
( 3 ) ( 1 )
( 6 ) ( 2 )
( 2 ) ( 4 ) ( 6 )
( 7 ) ( 4 )
( 8 ) ( 3 )
3. 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.
a. If n is fixed as 5, is this function one-to-one?
b. List five numbers that have the exact same image.
4. Find
5. Find
6. Find
7. Find as
8. Prove that is a continuous function.
9. Find derivatives of the following functions using differentiation rules. (Do not use the definition of a derivative!)
a.
b.
c.
d.
e.
f.
g.
h.
10.
11. 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)
12.
13.
f
g
o
x
x
f
1
)
(
=
0
1
x
=-
R
x
Î
g
f
o
