MATHEMATICS

.
lnx means the natural logarithm of x , so with base e ≈ 2 72 .
1. Simplify the following expressions ( , 0) x y > :
(a)
2 24
4
( ) ( ) ( )
x y y x
x y
– ¹ – (b)
1
6 3 2
6 3 4
2( )
64
x y
x y
–
– (c) 98
2 28 3 7 +
2. Factor completely:
(a) 4 2
x x – 4,25 1 , (b) p q p p pq ++ – + + + ( 1)( 1) 1
3. Solve the following equations:
(a) 2 ( 3 3)( 3) 1 ( 3) xx x x – + – =× –
(b) ( ) 3 log 2 1 2 x + =
(c) 2 5 25 3 + -= x x
4. Solve the following inequalities:
(a) 1
3 28 0 y y – – < (b) 0
2
x xe
x
£ – (c) 2 36 5 x xx <-£
5. Differentiate the following functions, simplify and factor if possible:
(a) ( ) x f x xe = (b) ln(1 )
1
x
y
x
+ = +
6. Let ( ) ln( ) x x fx e e- = +
(a) Find the domain of f (i.e. find all values x for which f x( ) is defined).
(b) Find the derivative f x'( ).
(c) Find the extreme values of f and classify (minimum/maximum).
7. Let 2 y fx x x = =- – () 4
(a) Find the domain of f (i.e. find all values x for which f x( ) is defined).
(b) Find the derivative f x'( ) and simplify.
(c) Find the equation of the tangent line at the graph of f with slope 1.
(d) Find the extreme values of f and classify (minimum/maximum and local/global).

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