Finding stationary points on a curve
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Transcript Finding stationary points on a curve
Differentiation 2.
LO: to be able to find stationary points on a curve using differentiation.
dy
d2y
Find dx and dx 2
a.
b.
c.
d2y
4
2
dx
y 2 x 3x 1
dy
4x 3
dx
y 2 x 5x 2 x 3
dy
6 x 2 10 x 2
dx
2
3
2
y x3 1
d.
y 2x
e.
3
y
x
f.
in the following:
1
2
x 1
y 3
x
dy
3x 2
dx
1
dy
x 2
dx
dy
3 x 2
dx
dy
2 x 3 3x 4
dx
d2y
12 x 10
2
dx
d2y
6x
2
dx
d2y
1 32
x
dx 2
2
d2y
3
6
x
dx 2
d2y
4
5
6
x
12
x
dx 2
Differentiation 2.
LO: to be able to find stationary points on a curve using differentiation.
Finding the turning points of the curve
y x3 6 x 2 9 x 1
What is the gradient at a stationary point??
dy
0
dx
dy
3 x 2 12 x 9
dx
3x 2 12 x 9 0
3
x2 4x2 3 0
( x 3)( x 1) 0
So, x = 1 or x = 3
Put into original equation to get y-coordinates:
Stationary points are: (1,5) and (3,1)
Differentiation 2.
LO: to be able to find stationary points on a curve using differentiation.
Sometimes we need to determine whether a stationary point is a
minimum point or a maximum point, without sketching a graph.
Maximum point
If
d2y
dx 2
is < 0, it is a maximum point.
If
d2y
dx 2
is > 0, it is a minimum point.
Minimum point
2
d y
6 x 12
2
dx
At x = 1
At x = 3
d2y
18
2
dx
d2y
6
2
dx
maximum
mimimum
Differentiation 2.
LO: to be able to find stationary points on a curve using differentiation.
Find the stationary points on the curve y x x x 2
whether they are minimum points or maximum points.
3
and determine
d2y
6x 2
2
dx
dy
3x 2 2 x 1
dx
At stationary points:
2
dy
0
dx
3x 2 2 x 1 0
(3x 1)( x 1) 0
49
1
y
x
27
3
x 1
y 3
When x = 1/3,
d2y
1
6
24
2
dx
3
MINIMUM
When x = -1,
d2y
6 1 2 4
2
dx
MAXIMUM
There is a minimum point at (1/3, 49/27)
and a maximum point at (-1,3).
Differentiation 2.
LO: to be able to find stationary points on a curve using differentiation.
Find the turning values of each function and state whether the value is a
maximum or minimum.
a) y 2 x 3 9 x 2 2
3
2
d ) y x 2x x 7
b) y x 3 3x 2 10
c) y
1 3 1 2
x x 2x
3
2
e) y x
1
x
The curve y x ax bx 11 has a stationary point at (3, -70).
a) Use the fact that the curve passes through (3,-70) (i.e. y = -70 when x = 3) to
find an equation involving a and b.
b) Find dy and use the fact that dy 0 when x=3 to find another equation
dx
dx
involving a and b.
c) Solve these two equations to find a and b.
d) Find d y and use this to determine whether (3, -70) is a local maximum or
dx
minimum.
3
2
2
2