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Bài tập CALCULUS 59
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Created by T. Madas
Question 206
(****)
y
y = 2 x3 + 3x 2 − 11x − 6
O
Q
P
R
x
The figure above shows the curve with equation
y = 2 x3 + 3 x 2 − 11x − 6 .
The curve crosses the x axis at the points P , Q and R ( 2,0 ) .
The tangent to the curve at R is the straight line L1 .
a) Find an equation of L1 .
The normal to the curve at P is the straight line L2 .
The point S is the point of intersection between L1 and L2 .
b) Show that
PSR = 90° .
SYN-M , y = 25 x − 50
Created by T. Madas
Created by T. Madas
Question 207
(****)
A
4r
D
O θ
R
3r
C
B
The figure above shows a circular sector OAB of radius 4r subtending an angle θ
radians at the centre O . Another circular sector OCD of radius 3r also subtending an
angle θ radians at the centre O is removed from the first sector leaving the shaded
region R .
It is given that R has an area of 50 square units.
a) Show that the perimeter P , of the region R , is given by
P = 2r +
100
.
r
b) Given that the value of r can vary, …
i. … find an exact value of r for which P is stationary.
ii. … show clearly that the value of r found above gives the minimum
iii value for P .
c) Calculate the minimum value of P .
C2L , r = 5 2 ≈ 7.07 , Pmin = 20 2 ≈ 28.28
Created by T. Madas
Created by T. Madas
Question 208
(****)
y
O
x2 4
y=
−
2 x
P
x
The figure above shows the curve C with equation
y=
x2 4
− , x ≠ 0.
2 x
The curve crosses the x axis at the point P .
The straight line L is the normal to C at P .
a) Find …
i. … the coordinates of P .
ii. … an equation of L .
b) Show that L does not meet C again.
C1Y , P ( 2,0 ) , x + 3 y = 2
Created by T. Madas
