Final Examination
201-942-DW
Engineering Mathematics II
Winter 2008
1. (15 Marks) Evaluate the following limits.
a)
2
3
2
3 10
lim
4
x
x x
x x
→
−
=
−
+
b)
2
4 3
4
3 2 1
lim
2
6
x
x x
x x
→∞
+ −
=
−
−
c)
5
4 3
lim 5
x
x
x
→
+ −
=
−
2. (15 Marks) Find the first derivative of the following functions. Do not simplify your answers.
a)
2
tan(5 )
( )
x
f x
e=
b)
( )
4
ln
( ) cos 3
x
f x x
x
= +
c)
( )
2
3
6
( ) ln 3
x
f x x
−
=
−
3. (6 Marks) Given
(
)
2
( ) 1 arctan
f x x x
= + ⋅
.
a) Find
'( )
f x
and simplify. b) Find
"( )
f x
.
4. (5 Marks) Given
2 4 2
4 1
x y x y
+ = −
. Find
dy
dx
.
5. (8 Marks) Given the function
( ) 3
f x x
= +
.
a) Find equations of the tangent and normal lines to the graph of
( )
f x
at the point (1, 2).
b) Write the linear approximation for the function
( )
f x
at
1
x
=
and use it to
approximate
3.96
.
6.
(6 Marks) Given the function
4 3
1
2 54
4
( )
f x x
x
− +
=
. Find
a)
the intervals where
( )
f x
is increasing and where it is decreasing
b)
all local (relative) maxima and local (relative)minima
c)
the intervals where
( )
f x
is concave upward and where it is concave downward
d)
all points of inflection
Using this information sketch the graph of
( )
f x
. Clearly label all the points found.
7.
(5 Marks) The strength
S
of a beam with a rectangular cross section is directly proportional to the
product of its width
w
and the square of its height
h
2
( )
S kwh
=
. Find the dimensions of the
strongest beam that can be cut from a round log 30 cm in diameter.
8.
(20 Marks) Evaluate the following integrals:
a)
3
2
x x
dx
x
− +
∫
b)
3 1
x dx
−
∫
c) 5
sin cos
x x dx
∫
d)
3
4
1x
x e dx
−
∫
9.
(10 Marks)
a)
Find the area of the region bounded by the graphs of
2
2
y x
= −
and
2
y x
=
.
b)
Find the coordinates (
x
and
y
) of the center of mass of a thin plate covering the region from a).
10.
(5 Marks) Use the
disk method
to find the volume of the solid generated when the region
enclosed by the curves
y x
=
,
0
y
=
and
4
x
=
is revolved about the
x
-axis
.
11.
(5 Marks) Use the
shell method
to find the volume of the solid generated when the region
enclosed by the graphs of the functions
2
y x
=
and
3
y x
=
is revolved about the
y
-axis
.
