7.

Sol: (1) (a) Even function: The function ( )f x is said to be even if and only if ( )f x
is a real –valued function of a real variable x , and ( ) ( ).f x f x 
Example of Even function
The function 2( ) 3 4f x x   is an even function as:
 
 
2
2
( ) 3 4
( ) 3 4
( ) ( )
f x x
f x x
f x f x
    
    
 

So, ( ) ( )f x f x 
Sol: (1) (b) Work: When a force acts to move an object, we say that Work was
done on the object by the Force.
.W F s

Force vector applied to the object/system.
Displacement vector.
F
s



Work is usually defined in terms of dot product because it is only the
component of the force along the direction of motion of the objet.
Sol: (1) (c) Decreasing function: A function ( )f x decreases on an interval I if
( ) ( )f a f b for all ,b a where , .a b I If ( ) ( )f b f a for all ,b a the function is said to
be strictly decreasing.
If the derivative '( )f x of a continuous function ( )f x satisfies '( ) 0f x  on
an open interval ( , ),a b then ( )f x is decreasing on ( , ).a b However the function may
decrease on an interval without having a derivative defined at all points. For
example, the function
1
3x is decreasing everywhere, including the origin 0,x 
despite the fact that the derivative is not defined at that point.
Sol: (2) (a)    2
0
( ) lim 3 5 csc
x
f x x x x

  

   ( ) 0 0 5 csc 0
( )
f x
f x
  
 

Function is undefined, so limit does not exist.
Sol: (2) (b)
2
( ) lim tan
x
f x x




( ) tan
2
( )
f x
f x
 
  
 
 

Function is undefined, so limit does not exist.
Sol: (2) (c)
sin
( ) lim
x
x
f x
x


sin
( )
( ) 0
f x
f x




Sol: (3) The function g is increasing and concave up on the interval  3 .
To draw the graph,
From the graph,
 
3
lim 0
x
g x


Sol: (4) The function 4( ) 1f x x x  
To find the x-intercepts by making ( ) 0f x  and solve the equation

    
4 1 0
0.72 1.22 0.248 1.03 0.248 1.03 0
x x
x x x i x i
  
      

So
1
2
3
4
0.72
1.22
0.248 1.03
0.248 1.03
x
x
x i
x i

 
 
 
There are two real values, so the interval is  1.22,0.72 .
Sol:...