Bài tập CALCULUS 18

Created by T. Madas
Question 72 (***)

y
S

P

O

Q

R

x

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

y = 3x 2 − 8 x + 5 .
The curve crosses the x axis at the points P and Q .

a) Find the coordinates of P and Q .
b) Show clearly that



2

3 x 2 − 8 x + 5 dx = 0 .

1

The point R lies on the x axis where x = 2 and the point S lies on C so that RS is
parallel to the y axis.

c) Find the area of the shaded region.

( )

P (1,0 ) , Q 5 , 0
3

Created by T. Madas

8
27


Created by T. Madas
Question 73

(***)

A cubic curve has equation
f ( x ) = 2 x3 − 7 x 2 + 6 x + 1 .
The point P ( 2,1) lies on the curve.

a) Find an equation of the tangent to the curve at P .
The point Q lies on the curve so that the tangent to the curve at Q is parallel to the
tangent to the curve at P .

b) Determine the x coordinate of Q .
y = 2 x − 3 , xQ = 1
3

Created by T. Madas


Created by T. Madas
Question 74

(***)

y
y = x+4
B
A

y = x2 − 3x + 7
x

O

The figure above shows a curve C and a straight line L with respective equations

y = x 2 − 3x + 7 and y = x + 4 .
The curve and the straight line meet at the points A and B .

a) Find the coordinates of A and B .
b) Find the exact area of the finite region bounded by C and L , shown shaded in
the figure above.

A (1,5 ) , B ( 3, 7 ) , area = 4
3

Created by T. Madas