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Bài tập CALCULUS 21
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Created by T. Madas
Question 82
(***)
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.
C1D , f ( x ) =
Created by T. Madas
4
−2 ,
x
( 2,0 )
Created by T. Madas
Question 83
(***)
The total cost C , in £ , for a certain car journey, is modelled by
C=
200 2V
+
, V > 30 ,
V
25
where V is the average speed in miles per hour.
a) Find the value of V for which C is stationary.
b) Justify that this value of V minimizes C .
c) Hence determine the minimum total cost of the journey.
C2J , V = 50 , £8
Created by T. Madas
Created by T. Madas
Question 84
(***)
x
A
y B
G
F
1c
E
x
y C
D
The figure above shows a clothes design consisting of two identical rectangles attached
to each of the straight sides of a circular sector of radius x cm .
The rectangles measure x cm by y cm and the circular sector subtends an angle of
one radian at the centre.
The perimeter of the design is 40 cm .
a) Show that the area of the design, A cm 2 , is given by
A = 20 x − x 2 .
b) Determine by differentiation the value of x for which A is stationary.
c) Show that the value of x found in part (b) gives the maximum value for A .
d) Find the maximum area of the design.
SYN-H , x = 10 , Amax = 100
Created by T. Madas
