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Bài tập CALCULUS 11
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Created by T. Madas
Question 42
(***)
A curve C has the following equation
f ( x ) = 4x x −
25 x 2
, x≥0.
16
a) Find a simplified expression for f ′ ( x ) .
b) Determine an equation of the tangent to the curve at the point where x = 4 ,
giving the answer in the form ax + by = c , where a , b and c are integers.
1
C1I , f ′ ( x ) = 6 x 2 − 25 x , x + 2 y = 18
8
Question 43
(***)
The curve C has equation
y = x3 − 6 x 2 + 12 x − 5 .
Find the coordinates of the stationary point of C and use a clear method to determine
its nature.
C2M , point of inflexion at
Created by T. Madas
( 2,3)
Created by T. Madas
Question 44
(***)
f ( x) =
(
5 x 3x 2 − 2
x
) , x >0.
Show clearly that
5
f ( x ) dx = P x + Qx 2 + C ,
where P and Q are integers to be found, and C is an arbitrary constant.
P = −20 , Q = 6
Question 45
(***)
The point P ( 3,0 ) lies on the curve C whose gradient function is given by
dy
= 3 x 2 − 14 x + 12 .
dx
a) Find an equation of the tangent to C at the point P .
The point Q lies on C , so that the tangent at Q is parallel to the tangent at P .
b) Find the x coordinate of Q .
y = 9 − 3x , x = 5
3
Created by T. Madas
Created by T. Madas
Question 46
(***)
y
y = −4 x 2 + 24 x − 20
x
O
A
B
2
y = x − 6x + 5
The figure above shows the graph of the curves with equations
y = −4 x 2 + 24 x − 20 and
y = x2 − 6 x + 5 .
The two curves intersect each other at the points A and B .
The finite region R bounded by the two curves is shown shaded in the figure.
Find the exact area of R .
SYN-D , 160
3
Created by T. Madas
