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Bài tập CALCULUS 28
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Created by T. Madas
Question 106
(***+)
The curve C has equation
y = x3 − 9 x 2 + 24 x − 19 , x ∈ » .
a) Show that the tangent to C at the point P , where x = 1 , has gradient 9 .
b) Find the coordinates of another point Q on C at which the tangent also has
gradient 9 .
The normal to C at Q meets the coordinate axes at the points A and B .
c) Show further that the approximate area of the triangle OAB , where O is the
origin, is 11 square units.
Q ( 5,1)
Created by T. Madas
Created by T. Madas
Question 107
(***+)
y
y = x2 − 6 x + 5
C ( 7,12 )
O A
B
x
The diagram above shows the curve with equation
y = x2 − 6 x + 5 .
The point C ( 7,12 ) lies on the curve while A and B are the points of intersection of
the curve and the x axis.
Find the exact area of the shaded region, bounded by the curve, the straight line
segment AC and the x axis.
C2M , area = 76
3
Created by T. Madas
Created by T. Madas
Question 108
(***+)
1
y=
x 2 (3 x 2 + 1)
x2
, x >0.
Show clearly that
4
y dx = 15 .
1
proof
Question 109
(***+)
The gradient of every point on the curve C , with equation y = f ( x ) satisfies
f ′ ( x ) = 3x 2 − 4 x + k ,
where k is a non zero constant.
The points P ( 0, −3) and Q ( 2,7 ) both lie on C .
Find an equation for C .
y = x3 − 2 x 2 + 5 x − 3
Created by T. Madas
