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Bài tập CALCULUS 13
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Created by T. Madas
Question 51
(***)
y
A
y = 10 − 4 x − x 2
B
y = 6− x
x
O
The figure above shows a quadratic curve and a straight line with respective equations
y = 10 − 4 x − x 2
and
y = 6− x.
The points A and B , are the points of intersection between the quadratic curve and the
straight line.
Calculate the exact area of the finite region bounded by the quadratic curve and the
straight line, shown shaded in the above figure.
MP1-K , area = 125
6
Created by T. Madas
Created by T. Madas
Question 52
(***)
The gradient function of the curve with equation
2
y = 2( x + a) ,
where a is a non zero constant, is given by
dy
= 4 x + 10 .
dx
Determine the value of a .
MP1-B , a = 5
2
Question 53
(***)
The point P ( 4,9 ) lies on the curve with equation y = f ( x ) , whose gradient function
is given by
f ′( x) = 1+
2
, x >0.
x
Find an equation for f ( x ) .
f ( x) = x + 4 x − 3
Created by T. Madas
Created by T. Madas
Question 54
(***)
h
x
5x
The figure above shows a solid brick, in the shape of a cuboid, measuring 5x cm by
x cm by h cm . The total surface area of the brick is 720 cm 2 .
a) Show that the volume of the brick, V cm3 , is given by
V = 300 x −
25 3
x .
6
b) Find the value of x for which V is stationary.
c) Calculate the maximum value for V , fully justifying the fact that it is indeed
the maximum value.
C2B , x = 2 6 ≈ 4.90 , Vmax = 400 6 ≈ 980
Created by T. Madas
