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Bài tập CALCULUS 61
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Created by T. Madas
Question 212
(****)
y
x
The figure above shows a triangular prism whose triangular faces are parallel to each
other and are in the shape of equilateral triangles of side length x cm .
The length of the prism is y .
a) Given that total surface area of the prism is exactly 54 3 cm 2 , show clearly
that the volume of the prism, V cm3 , is given by
V = 27 x − 1 x3 .
2
8
b) Find the maximum value of V , fully justifying the fact that it is indeed the
maximum value.
c) Determine the value of y when V takes this maximum value.
C2O , Vmax = 27 , y = 2 3
Created by T. Madas
Created by T. Madas
Question 213
(****)
y
x
O R
P
y = 4 x − 3x − 3
The figure above shows part of the curve with equation
y = 4 x − 3x − 3 , x > 0
a) Show that an equation of the tangent to the curve at the point P , where x = 4 ,
is given by
y = 1 − 2x .
The finite region R is bounded by the curve, the tangent to the curve at P and the
coordinate axes.
b) Determine the exact area of R .
MP1-P , area = 29
12
Created by T. Madas
Created by T. Madas
Question 214
(****)
The straight line with equation
y = 2x + c
is a tangent to the curve with equation
y = x2 + 6 x + 7 .
Without using the discriminant, determine the value of the constant c and find the
point of contact between the tangent and the curve.
c=3 ,
Created by T. Madas
( −2, −1)
