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Bài tập CALCULUS 17
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Created by T. Madas
Question 68
(***)
A curve with equation y = f ( x ) passes through the point ( 2,3) .
The gradient function of the curve is given by
f ′ ( x ) = ( x − 3)( 3x − 1) .
a) Find an equation of the curve, giving the answer as a polynomial in its simplest
form.
b) Show clearly that
2
f ( x ) ≡ ( x + k )( x − 3) ,
where k is a constant to be found.
c) Sketch the graph of f ( x ) .
The sketch must show the coordinates of any points where the graph of f ( x )
meets the coordinate axes.
C1W , f ( x ) ≡ x3 − 5 x 2 + 3x + 9 , k = 1
Created by T. Madas
Created by T. Madas
Question 69
(***)
h
x
x
The figure above shows the design of a large water tank in the shape of a cuboid with a
square base and no top.
The square base is of length x metres and its height is h metres.
It is given that the volume of the tank is 500 m3 .
a) Show that the surface area of the tank, A m 2 , is given by
A = x2 +
2000
.
x
b) Find the value of x for which A is stationary.
c) Find the minimum value for A , fully justifying the fact that it is the minimum.
MP1-C , x = 10 , Amin = 300
Created by T. Madas
Created by T. Madas
Question 70 (***)
f ( x ) = 2 x 2 + 3 x + k , where k is a constant.
Find the value of k , given that
3
1
f ( x ) dx =
4
.
3
k = −14
Question 71
(***)
The point P ( −1, −1) lies on the curve C , whose gradient function is given by
dy 5 x3 − 6
=
, x ≠ 0.
dx
x3
Find an equation for C .
y = 5x +
Created by T. Madas
3
x2
+1
