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Bài tập CALCULUS 54
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Created by T. Madas
Question 191
(****)
y
y = f ( x)
O
P ( −1,0 )
Q
x
The figure above shows a curve with equation y = f ( x ) .
The curve meets the x axis at the points P ( −1, 0 ) and Q , and its gradient function is
given by
f ′( x) =
8 x3 − 1
x2
, x ≠ 0.
a) Find an equation of the tangent to the curve at P .
b) Find an expression for f ′′ ( x ) .
c) Determine …
i. … an equation of the curve.
ii. … the coordinates of Q .
1
C1A , y = −9 x − 9 , f ′′ ( x ) = 8 + 2 x −3 , y = 4 x 2 + − 3 , Q 1 ,0
2
x
( )
Created by T. Madas
Created by T. Madas
Question 192
(****)
y
A
M
y = x (6 − x)
O
x
The figure above shows the curve C with equation
y = x (6 − x) , x ∈ » .
The point M is the maximum point of C and the point A has coordinates ( 0,12 ) .
Find the exact area of the shaded region, bounded by the curve, the y axis and the
straight line segment from A to M .
SYN-G , area = 27
2
Created by T. Madas
Created by T. Madas
Question 193
(****)
y
2x
The figure above shows the design of a window which is the shape of a semicircle
attached to rectangle. The diameter of the semicircle is 2x m and is attached to one
side of the rectangle also measuring 2x m . The other side of the rectangle is y m .
The perimeter of the window is 6 m .
a) Show that the total area of the window, A m 2 , is given by
A = 6 x − 1 ( 4 + π ) x2 .
2
b) Given that the measurements of the window are such so that A is maximum,
show by a method involving differentiation that this maximum value of A is
18
.
4 +π
MP1-Y , proof
Created by T. Madas
