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Bài tập CALCULUS 12
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Created by T. Madas
Question 47
(***)
h
x
2x
The figure above shows the design of a fruit juice carton with capacity of 1000 cm3 .
The design of the carton is that of a closed cuboid whose base measures x cm by
2 x cm , and its height is h cm .
a) Show that the surface area of the carton, A cm 2 , is given by
A = 4 x2 +
3000
.
x
b) Find the value of x for which A is stationary.
c) Calculate the minimum value for A , justifying fully the fact that it is indeed the
minimum value of A .
MP1-P , x = 3 375 ≈ 7.21 , Amin ≈ 624
Created by T. Madas
Created by T. Madas
Question 48
(***)
The point P , where x = 3 , lies on the curve C whose equation is
y = 1 x3 − 3 x 2 + 3 x + 2 .
3
a) Find an equation of the tangent to C at the point P .
b) Show that there is no other point on C where the gradient is the same as the
gradient at P .
y = 11 − 6 x
Question 49
(***)
5
x6 +1
y= 3
, x >0.
x
Show clearly that
1
a
y dx = ax a + 1
a x +C ,
where a is a rational constant to be found, and C is an arbitrary constant.
a = 2 or 3
3
2
Created by T. Madas
Created by T. Madas
Question 50
(***)
The cubic equation C passes through the origin O and its gradient function is
dy
= 6 x 2 − 6 x − 20 .
dx
a) Show clearly that the equation of C can be written as
y = x ( 2 x + a )( x + b ) ,
where a and b are constants.
b) Sketch the graph of C , indicating clearly the coordinates of the points where
the graph meets the coordinate axes.
C1G , a = 5 , b = −4
Created by T. Madas
