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Bài tập CALCULUS 24
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Created by T. Madas
Question 91
(***+)
y
y = 4 ( x − 2)
2
A
y = 2 x 2 − 9 x + 16
B
O
x
The figure above shows the graph of the curves with equations
y = 4 ( x − 2)
2
and
y = 2 x 2 − 9 x + 16 .
The curves meet each other at the points A and B .
a) Determine the coordinates of A and B .
b) Find the exact area of the finite region bounded by the two curves, shown
shaded in the above figure.
( )
MP1-E , A ( 0,16 ) B 7 ,9 , 343
2
24
Created by T. Madas
Created by T. Madas
Question 92
(***+)
3x2 − 2 )
(
f ( x) =
2
x2
, x ≠ 0.
Show clearly that
2
f ( x ) dx = 11 .
1
proof
Question 93
(***+)
The curve C has equation
y=
3−8 x
, x>0
x
Show clearly that
d2y
dx
2
=
(
6 1− x
x
3
).
proof
Created by T. Madas
Created by T. Madas
Question 94
(***+)
r
h
A pencil holder is in the shape of a right circular cylinder, which is open at one of its
circular ends.
The cylinder has radius r cm and height h cm and the total surface area of the
cylinder, including its base, is 360 cm 2 .
a) Show that the volume, V cm3 , of the cylinder is given by
V = 180r − 1 π r 3 .
2
b) Determine by differentiation the value of r for which V has a stationary value.
c) Show that the value of r found in part (b) gives the maximum value for V .
d) Calculate, to the nearest cm3 , the maximum volume of the pencil holder.
r=
Created by T. Madas
120
π
≈ 6.18 , Vmax ≈ 742
