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Bài tập CALCULUS 32
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Created by T. Madas
Question 119
(***+) non calculator
The curve C has equation
y = 4x − x , x ≥ 0 .
Find the coordinates of the stationary point of C , and determine its nature.
(
min 1 , − 1
64 16
Question 120
)
(***+)
1
x − 12 − 3 , x > 0 .
f ( x ) = x 2 − 4
Show clearly that
3
f ( x ) dx = P x + Qx + Rx 2 + C ,
where P , Q and R are integers to be found, and C is an arbitrary constant.
P = −8 , Q = 13 , R = −2
Created by T. Madas
Created by T. Madas
Question 121
(***+)
f ( x ) ≡ x3 − 2 x 2 − x − 6 , x ∈ » .
a) Use the factor theorem to show that ( x − 3) is a factor of f ( x ) .
b) Hence express f ( x ) as the product of a linear and a quadratic factor.
The curve C has equation
y = 3x 4 − 8 x3 − 6 x 2 − 72 x + 240 .
c) Show that C has a single stationary point, and determine its coordinates and its
nature.
(
)
C2V , f ( x ) ≡ ( x − 3 ) x 2 + x + 2 , min ( 3, −3)
Created by T. Madas
Created by T. Madas
Question 122
(***+)
y
y = x 2 − 8 x + 18
Q ( 8,18)
P ( 2,6 )
x
O
The figure above shows the parabola with equation
y = x 2 − 8 x + 18 , x ∈ » .
The points P ( 3,3) and Q ( 6,6 ) both lie on the parabola.
Find the exact of the shaded region, bounded by the curve and the straight line segment
between P and Q .
SYN-R , area = 36
Created by T. Madas
