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Bài tập CALCULUS 42
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Created by T. Madas
Question 155
(***+)
y
y = 6 x 2 − 4 x3
A1
O
k
3
2
x
A2
The figure above shows the graph of the curve with equation
y = 6 x 2 − 4 x3 , x ∈ » .
( )
The curve meets x axis at the origin O and at the point 3 ,0 .
2
The point ( k ,0 ) , k > 3 is such so that, the area A1 of the region between the curve
2
and the x axis for which 0 ≤ x ≤ 3 , is equal to the area A2 of the region between the
2
3
curve and the x axis for which ≤ x ≤ k .
2
Determine the value of k .
MP1-F , k = 2
Created by T. Madas
Created by T. Madas
Question 156
(***+)
The curve C has equation
y = 5x +
4
−3, x ≠ 0
x
Show that the straight line with equation
y = 4x + 1
is a tangent to C , and find the coordinates of the point of tangency.
SYN-H ,
Created by T. Madas
( 2,9 )
Created by T. Madas
Question 157
(***+)
h
(l − 2)
l
The figure above shows 12 rigid rods, joined together to form the framework of a
storage container, which in the shape of a cuboid.
Each of the four upright rods has height h m . Each of the longer horizontal rods has
length l m and each of the shorter horizontal rods have length ( l − 2 ) m .
a) Given that the total length of the 12 rods is 36 m show that the volume, V m3 ,
of the container satisfies
V = −2l 3 + 15l 2 − 22l .
b) Find, correct to 3 decimal places, the value of l which make V stationary.
c) Justify that the value of l found in part (b) maximizes the value of V , and find
this maximum value of V , correct to the nearest m3 .
d) State the three measurements of the container when its volume is maximum.
l = 4.107 , Vmax ≈ 24 , 4.11× 2.11× 2.79
Created by T. Madas
