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Bài tập CALCULUS 36
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Created by T. Madas
Question 135
(***+)
y
L
N
R
C
O
M
A
x
The figure above shows the graph of the curve C with equation
y = 6x − x2 , x ∈ » .
The curve meets the x axis at the origin O and at the point A . The straight line L is
the tangent to C at A .
a) Find an equation of L .
The point M is the maximum point of C . The point N lies on L so that MN is
parallel to the y axis. The finite region R , shown shaded in the figure above, is
bounded by C , L and the straight line segment MN .
b) Determine the area of R .
MP1-I , y = 36 − 6 x , 9
Created by T. Madas
Created by T. Madas
Question 136
(***+)
The curve C has equation
y=
( x + 4 )2 ,
x
x >0.
a) Show that the gradient function of C is
dy 3 12
−1
−3
= x + 4 x 2 − 8x 2 .
dx 2
The point P lies on C where x = 4 .
The straight line L is the tangent to C at the point P .
b) Find an equation of L .
c) Find the area of the triangle OAB , where A and B are the points where L
crosses the coordinate axes, and O is the origin.
y = 4 x + 16 , area = 32
Created by T. Madas
Created by T. Madas
Question 137
(***+)
The figure below shows a large tank in the shape of a cuboid with a rectangular base
and no top.
Two of the vertical opposite faces of the cuboid are square and the height of the cuboid
is x metres.
x
a) Given that the surface area of the tank is 54 m 2 , show that the capacity of the
tank, V m3 , is given by
V = 18 x − 2 x3 .
3
b) Find the maximum value for V , fully justifying the fact that it is indeed the
maximum value.
Vmax = 36
Created by T. Madas
