Bài tập Toán INTEGRATION 09

Created by T. Madas
Question 19
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 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


Created by T. Madas
Question 20
y

R
C

Q

P
O

x

The figure above shows the curve C which meets the coordinates axes at the points P ,
Q and R .
Given the gradient function of C is given by

f ′( x) = 3 − 4x ,
and that f (1) = 2 f ( 2 ) , determine the coordinates of P , Q and R .

( )

P ( −1, 0 ) , P 5 , 0 , R ( 0,5 )
2

Created by T. Madas


Created by T. Madas
Question 21
The curve C with equation y = f ( x ) satisfies
f ′( x) = −

4
x2

, x ≠ 0.

a) Given that f (1) = 2 , find an expression for f ( x ) .
b) Sketch the graph of f ( x ) , indicating clearly the asymptotes of the curve and
the coordinates of any points where the curve crosses the coordinate axes.
f ( x) =

4
−2 ,
x

( 2,0 )

Question 22
1
−1

f ( x ) =  x 2 − 4 
 x 2 − 3  , x > 0 .




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