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Bài tập CALCULUS 65
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Created by T. Madas
Question 224
(****)
f ( x ) = x3 + x 2 − x + 15 , x ∈ »
a) Show that ( x + 3) is a factor of f ( x ) .
The figure below shows the curve C with equation y = f ( x ) .
y
A
B
y = f ( x)
R
O
x
The points A and B are stationary points of C .
b) Find the exact coordinates of A and B .
The finite region R is bounded by the curve, the x axis and the straight line segment
OA , where O is the origin.
c) Determine the exact area R .
(
)
C2G , A ( −1,16 ) , B 1 , 400 , area = 92
3 27
3
Created by T. Madas
Created by T. Madas
Question 225
(****)
The curve C has equation
y=
1
2 x2
+
4
3x3
, x>0.
Show clearly that
x2
d2y
dx
2
+ 6x
dy
+ 6y = 0 .
dx
C1T , proof
Question 226
(****)
f ( x) ≡ x +
8
x2
, x>0.
Find the range of values of x , for which f ( x ) is increasing.
x>4
Created by T. Madas
Created by T. Madas
Question 227
(****)
y
L
R
O
y = 1 x2 − 9 x + 7
2
2
P
Q
x
The diagram above shows the quadratic curve C with equation
y = 1 x2 − 9 x + 7 .
2
2
The curve crosses the x axis at the points P and Q , and the y axis at the point R .
The line L is the tangent to C at the point P .
a) Find an equation of L .
b) Find the exact area of the shaded region bounded by the tangent at P , the curve
and the y axis.
2 y + 5 x = 10 , area = 4
3
Created by T. Madas
