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Bài tập CALCULUS 39
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Created by T. Madas
Question 146
(***+)
x
x
x
θ
x
x
A wire of total length 60 cm is to be cut into two pieces. The first piece is bent to form
an equilateral triangle of side length x cm and the second piece is bent to form a
circular sector of radius x cm . The circular sector subtends an angle of θ radians at
the centre.
a) Show that
xθ = 60 − 5 x .
The total area of the two shapes is A cm 2 .
b) Show clearly that
A=
1
4
(
)
3 − 10 x 2 + 30 x .
c) Use differentiation to find the value of x for which A is stationary.
d) Find, correct to three significant figures, the maximum value of A , justifying
the fact that it is indeed the maximum value of A .
SYN-F , x ≈ 7.26 , Amax ≈ 109
Created by T. Madas
Created by T. Madas
Question 147
(***+)
y
2
y = 2x 3 − x
A
O
x
The figure above shows part of the curve with equation
2
y = 2x 3 − x , x ≥ 0 .
The curve meets the x axis at the point A .
a) Show that the coordinates of A are ( 8,0 ) .
b) Find the exact area of the finite region bounded by the curve and the x axis .
area = 32
5
Created by T. Madas
Created by T. Madas
Question 148
(***+) non calculator
y=
2 − 3x x
1
+
, x >0.
3x
2
a) Find an expression for
dy
.
dx
b) Show that the value of
d2y
5
where x = 4 , is
.
2
dx
96
dy
2
1 −1
= − x −2 − x 2
dx
3
2
Created by T. Madas
