Bài tập CALCULUS 19

Created by T. Madas
Question 75

(***)

The curve C has equation
y = 4 x5 − 1 , x ≥ 0
Show clearly that

4 x2

d2y
dx 2

− 15 y = k ,

where k is an integer to be found.
C1E , k = 15

Question 76

(***)
2

f ′ ( x ) = ( 3x − 1) .
Given that f ( 3) = 56 , find an expression for f ( x ) .
C1N , f ( x ) = 3x3 − 3x 2 + x − 1

Created by T. Madas

Created by T. Madas
Question 77

(***)

A
8x

6x

E

B

y
D

10 x

C

The figure above shows a pentagon ABCDE whose measurements, in cm , are given
in terms of x and y .

a) If the perimeter of the pentagon is 120 cm , show clearly that its area, A cm 2 ,
is given by
A = 600 x − 96 x 2 .

b) Use a method based on differentiation to calculate the maximum value for A ,
fully justifying the fact that it is indeed the maximum value.

SYN-S , Amax = 937.5

Created by T. Madas


Created by T. Madas
Question 78

(***)

y

B

y = x2 − 2 x + 4
A

R
y = 3x

O

P

x

The figure above shows the graph of the curve C with equation

y = x2 − 2 x + 4 , x ∈ »
intersected by the straight line L with equation

y = 3x , x ∈ » .
The curve meets the straight line at the points A and B .
The point P is located on the x axis so that the straight line segment BP is parallel to
the y axis.
The finite region R is bounded by C , L , BP and the x axis.
Show that the area of R , shown shaded in the figure, is 39 .
2
MP1-J , proof

Created by T. Madas