Bài tập CALCULUS 44

Created by T. Madas
Question 161

(***+)

y

Q (1,0 )

O

R

x

P

The figure above shows the graph of a cubic curve, which touches the x axis at the
point Q (1,0 ) .

a) Determine an equation for the cubic curve , given its gradient function is
dy
= 3 x 2 − 12 x + 9 .
dx
The cubic curve crosses the x axis and the y axis at the points R and P , respectively.

b) Determine the coordinates …
i. … of the point P .
ii. … of the point R .
C1I , y = x3 − 6 x 2 + 9 x − 4 , P ( 0, −4 ) , R ( 4,0 )

Created by T. Madas


Created by T. Madas
Question 162

(***+)

y

C

L
R
Q

P
x

O

The figure above shows a sketch of the curve C with equation

y = 2x2 , x ∈ » .

( )

The straight line L passes through the points P 1 , 1 and Q ( 0,1) , where the point
2 2
P lies on C .

The straight line L meets C again, at the point R .

a) Find the coordinates of R .
The tangents to C at the points P and R meet at the point T .

(

)

b) Show that the coordinates of T are − 1 , −1
4

R ( −1, 2 )

Created by T. Madas


Created by T. Madas
Question 163

(***+)

A cubic curve has equation

y = x3 − x 2 + 5 , x ∈ » .
The point P lies on the curve where x = 1 .
Show that the normal to the curve at P does not meet the curve again.
SYN-R , proof

Created by T. Madas