optimisation using derivatives i
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Optimisation Using Optimisation Using Derivatives IDerivatives I
Positive and negative values of the derivative as an indication of the points at which the function is increasing or decreasing
zero values of the derivative as an indication of stationary points
Greatest and least values of functions
relationship between the graph of a function and the graph of its derivative
methods of determining the nature of stationary points
greatest and least values of functions in a given interval
If the gradient is positive, the curve slopes upwards from left to If the gradient is positive, the curve slopes upwards from left to rightright
If the gradient is negative, the curve slopes downwards from left If the gradient is negative, the curve slopes downwards from left to rightto right
Page Page 501 Ex 21.1501 Ex 21.11-21-2 (orally)(orally)
State the values of x for which the curve is State the values of x for which the curve is (a) increasing(a) increasing(b) decreasing(b) decreasing(c) stationary(c) stationary
State the values of x for which the curve is State the values of x for which the curve is (a) increasing x<3, x>2(a) increasing x<3, x>2(b) decreasing(b) decreasing(c) stationary(c) stationary
State the values of x for which the curve is State the values of x for which the curve is (a) increasing x<3, x>2(a) increasing x<3, x>2(b) decreasing -3<x<2(b) decreasing -3<x<2(c) stationary(c) stationary
State the values of x for which the curve is State the values of x for which the curve is (a) increasing x<3, x>2(b) decreasing -3<x<2(c) stationary x=-3, x=2
Positive and negative values of the Positive and negative values of the derivative as an indication of the points derivative as an indication of the points
at which the function is increasing or at which the function is increasing or decreasingdecreasing
Consider the following functions and Consider the following functions and their derivatives:their derivatives:
y = xy = x22
y = xy = x33
y = xy = x33 - 2x - 2x22 – 2x + 1 – 2x + 1
Page Page 501 Ex 21.1501 Ex 21.133
Greatest and Least ValuesGreatest and Least Values
Find the greatest and least values of the Find the greatest and least values of the function y = xfunction y = x33 + 2x + 2x22 - 4x + 5 - 4x + 5
for -3 for -3 x x 2 2
Check bounds
When x = -3, y= 8 When x = 2, y = 13
Check stationary values
y = xy = x3 3 + 2x+ 2x22 - 4x + 5 - 4x + 5
y’ = 3xy’ = 3x2 2 + 4x – 4 = (3x-2)(x+2)+ 4x – 4 = (3x-2)(x+2)
S.P. when y’=0 S.P. when y’=0 x= x= ⅔⅔ or x = -2 or x = -2
When x = When x = ⅔⅔, y = , y = 3.5193.519 (3dp) When x= -2, y = (3dp) When x= -2, y = 1313
Greatest value is 13 ; Least value is 3.519 (3dp)Greatest value is 13 ; Least value is 3.519 (3dp)
y = xy = x33 + 2x + 2x22 - 4x + 5 - 4x + 5
Page Page 505 Ex 21.2505 Ex 21.211
Model : Find the stationary points on the curveModel : Find the stationary points on the curve y = x y = x33+2x+2x2 2 – 4x +2– 4x +2
2
0)2)(23(
0443
0
)2)(23(
443
242
32
2
2
23
xorx
xx
xx
whenoccurSP
xx
xx
xxxy
dxdy
dxdy
When x = ⅔, y = 0.519 (3dp)
When x = -2, y = 10
X = -2X = -2 X = X = ⅔⅔
dydy = = dxdx
When x = ⅔, y = 0.519 (3dp)
When x = -2, y = 10
X < -2X < -2 X = -2X = -2-2 < x < -2 < x <
⅔⅔ X = X = ⅔⅔ X > X > ⅔⅔
dydy = = dxdx
When x = ⅔, y = 0.519 (3dp)
When x = -2, y = 10
X < -2X < -2 X = -2X = -2-2 < x < -2 < x <
⅔⅔ X = X = ⅔⅔ X > X > ⅔⅔
00 00dydy = = dxdx
When x = ⅔, y = 0.519 (3dp)
When x = -2, y = 10
X < -2X < -2 X = -2X = -2-2 < x < -2 < x <
⅔⅔ X = X = ⅔⅔ X > X > ⅔⅔
++ 00 -- 00 ++dydy = = dxdx
When x = ⅔, y = 0.519 (3dp)
When x = -2, y = 10
X < -2X < -2 X = -2X = -2-2 < x < -2 < x <
⅔⅔ X = X = ⅔⅔ X > X > ⅔⅔
++ 00 -- 00 ++dydy = = dxdx
Max at (-2, 10) Min at (⅔ ,0.519)
Now check your answer on your GC
Read bottom of 509 - 510Read bottom of 509 - 510
Page Page 510 Ex 21.3510 Ex 21.31 a,d,e,f1 a,d,e,f4-6, 9, 34-6, 9, 3
The The SecondSecond Derivative Derivative
The derivative of is called the second The derivative of is called the second
derivative derivative
At a S.P.,At a S.P.,
if is positive then that point is a minimum if is positive then that point is a minimum
if is negative then that point is a maximum if is negative then that point is a maximum
dx
dy
2
2
dx
yd
2
2
dx
yd
2
2
dx
yd
Model Model Find the nature of the stationary Find the nature of the stationary pointspoints
on the curve y = on the curve y = xx33+2x+2x2 2 – 4x +2– 4x +2
2
0)2)(23(
0443
0
)2)(23(
443
242
32
2
2
23
xorx
xx
xx
whenoccurSP
xx
xx
xxxy
dxdy
dxdy
)10,2(max
,2
)519.0,(min
,
46
2
2
2
2
2
2
32
32
atimum
negxWhen
atimum
posxWhen
x
dx
yd
dx
yd
dx
yd
Model Model Draw the graph of y = 2 sin 3x for 0Draw the graph of y = 2 sin 3x for 0 < x < x < 2< 2ΠΠ
y = 2 sin 3xy = 2 sin 3xdy/dx = 6 cos 3xdy/dx = 6 cos 3xdd22y/dxy/dx22 = -18 sin 3x = -18 sin 3x
S.P. when dy/dx = 0 S.P. when dy/dx = 0 6 cos 3x = 06 cos 3x = 0 cos 3x = 0cos 3x = 0 3x = (2n+1)3x = (2n+1)ΠΠ/2/2 x = x = (2n+1)(2n+1)Π/Π/66 x = … x = … ΠΠ/6, 3/6, 3ΠΠ/6, 5/6, 5ΠΠ/6, 7/6, 7ΠΠ/6, …/6, …
xx dy/dy/dxdx
dd22y/dxy/dx2 2
(-18 sin 3x)(-18 sin 3x)
ΠΠ/6/6 00 --
ΠΠ/2/2 00 ++
55ΠΠ/6/6 00 --
77ΠΠ/6/6 00 ++
33ΠΠ/2/2 00 --
Max at (Max at (ΠΠ/6,2)/6,2)
Min at (Min at (ΠΠ/2,-2)/2,-2)
Max at (5Max at (5ΠΠ/6,2)/6,2)
Min at (7Min at (7ΠΠ/6,-2)/6,-2)
Max at (3Max at (3ΠΠ/2,2)/2,2)
1111ΠΠ/6/6 00 ++ Min at (11Min at (11ΠΠ/6,-2)/6,-2)
Check on GC
Page Page 510 Ex 21.3510 Ex 21.344
Repeat
using the second derivative to determine the nature of stationary points
+ Page + Page 518 Ex 21.4518 Ex 21.422
PerformanceFaith Honour
St Luke’s Anglican School
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