The presentation includes all the necessary material for review of Chapter 7.2 Stationary Points for A-level Pure Mathematics Cambridge International Examinations. It will be useful for students as the review materials and also for teachers as the teaching tips and presentations in the class.
The presentation was made by Oleksii Khlobystin (Alex) - Mathematics Teacher at Depu Foreign Language School, Chongqing City, China.
Personal website: www.visualcv.com/o-khlobystin
Personal email: o-khlobystin@yandex.comPresentation POWER POINT Chapter 7.2 Stationary Points, A-level Pure Mathematics CIE 9709
Made by Oleksii Khlobystin (Alex) - Mathematics Teacher at Depu Foreign Language School – Cambridge International Centre, Chongqing City, China.
Personal website: www.visualcv.com/o-khlobystin
Personal email: o-khlobystin@yandex.com
e.g.
where the gradient is zero
0dxdyThe stationary points of a curve are the points
The word local is usually omitted and the
points called maximum and minimum points.
A local
maximum
x
x
A local
minimum
3
x
y
3 2
x
9
x
e.g.1 Find the coordinates of the stationary
points on the curve xxxy93230dxdySolutio
n: xxxy9323dxdy9632xx0)32(32xx0)1)(3(3xxo
1, 5) 93127509632xxTip: Watch out for common
r3x1xyx3 272727yx1 )1(9)1(3)1(23)3(9)3(3)3(23The stationary points are (3, 27) and (
factors when finding stationary
points.
Exercise
sFind the coordinates of the stationary points
of the following functions542xxy1.
2. 1123223xxxy
s: 0420xdxdy2x15)2(4)2(22yx42xdxdy1.
Solution
Ans: St. pt. is ( 2, 1)
2. 1123223xxxy 21xxor 61yx 211)2(12)2(3)2(22 23 yx 12662xxdxdySolutio
0)2)(1(6xx
n: 0)2(602xxdxdy
Ans: St. pts. are ( 1, 6) and ( 2, 21 )
is negative
is positiveWe need to be able to determine the nature of
a stationary point ( whether it is a max or a
min ). There are several ways of doing this.
e.g.
On the left of
a maximum,
the gradient
On the right
of a
maximum,
the gradient
are0The opposite is true for a minimum0At the
Calculating the gradients on the left and right of
a stationary point tells us whether the point is a
So, for a max the gradients
max or a min.
On the left of
the max
max
On the right of
the max
e.g.2 Find the coordinates of the stationary point
of the curve . Is the point a max
or min? 142xxy
: 42xdxdy0420xdxdy 1)2(4)2(2y2x142xxy )1(
3y24)1(2dxdy
24)3(2dxdy00
We have 0 )3,2(is a min
Substitute in (1):
On the left of x = 2 e.g. at x =
On the right of x = 2 e.g. at x =
Solution
1,
3,
by 9632xxdxdy109323xxxye.g.3 Consider
At the max of 109323xxxy dxdy
but the gradient of the gradient is
negative.
the gradient is
0
Another method for determining the nature of
a stationary point.
The gradient
function is given
Another method for determining the
nature of a stationary point.
by 9632xxdxdy109323xxxye.g.3 Consider
of 109323xxxythe gradient of
“d 2 y by d x squared” 22dxyddxdy
the gradient is
positive.
The notation for the gradient of the gradient
The gradient
function is given
At the min
is
between the max and the min.109323xxxy 22dxydSolutio
n: 109323xxxyStationary
66x9632xxe.g.3 ( continued ) Find the stationary points
dxdy
points: 0dxdy09632xx0)32(32xx0)1)(3(3xx1x3xor
2nd derivative22dxyd
We now need to find the y-coordinates of the st.
pts.
on the curve and distinguish
is called the
at )5,1(3xAt , 22dxyd12661xAt , 22dxyd10931y109323xxxyTo distinguish between max and min we use
3x10)3(9)3(3)3(23 y 371x5126)3(6 max
the 2nd derivative, at the stationary points. 6622xdxyd
at )37,3(00min
SUMMAR
Y To find stationary points, solve the
equation 0dxdy0maximu
m0 minimu
derivative at the stationary pointsmin022dxydmax022dxyd
Determine the nature of the stationary
points
• either by finding the gradients on the
left and right of the stationary points
• or by finding the value of the 2nd
m
Exercise
sFind the coordinates of the stationary points
of the following functions, determine the
nature of each and sketch the functions. 2323xxy1.
2. 332xxy)2,0(is a
min.)2,2(is a
2323xxy
Ans. )0,1(is a
min.)4,1(is a
332xxy
Ans.
max.
max.
Chapter 7.2 Stationary Points-Alex.ppt
