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Bài tập CALCULUS 75
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Created by T. Madas
Question 254
(****+)
A curve C has equation
y = 2 x3 − 5 x 2 + a , x ∈ » ,
where a is a constant.
The tangent to C at the point where x = 2 and the normal to C at the point where
x = 1 , meet at the point Q .
Given that Q lies on the x axis, determine in any order …
a) … the value of a .
b) … the coordinates of Q .
( )
C1W , a = 8 , Q 7 ,0
3
3
Created by T. Madas
Created by T. Madas
Question 255
(****+)
The figure below shows the design of a window which is the shape of a semicircle
attached to rectangle.
y
2x
The diameter of the semicircle is 2 x metres and is attached to one side of the rectangle
also measuring 2 x meters. The other side of the rectangle is y metres.
The total area of the window is 2 m 2 .
a) Show that perimeter, P m , is given by
P=
1
2
(4 + π ) x + .
2
x
b) Determine by differentiation an exact value of x for which P is stationary.
[continues overleaf]
Created by T. Madas
Created by T. Madas
[continued from overleaf]
c) Show that the value of x found in part (b) gives the minimum value for P .
d) Show that when P takes a minimum value x = y .
x=
Created by T. Madas
2
≈ 0.748
π +4
