Y X The derivative Lecture 5 Handling a changing world Y X x 2 - x 1 y 2 - y 1 The derivative x 2 - x 1 y 2 - y 1 x y x x y y slope 1 2 1 2 x x f x x f x x x f x f slope ) ( ) ( ) ( ) ( 1 1 1 2 1 2 x 1 x 2 y 1 y 2 x x f x x f slope x ) ( ) ( lim 0 x x f x x f y x f dx dy x ) ( ) ( lim ) ( ' 0 The derivative describes the change in the slope of functions Aryabha ta (476- Bhaskara II (1114- 1185) The first Indian satellite
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The derivative Lecture 5 Handling a changing world x 2 -x 1 y 2 -y 1 The derivative x 2 -x 1 y 2 -y 1 x1x1 x2x2 y1y1 y2y2 The derivative describes the.
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Y
X
The derivative
Lecture 5Handling a changing world
Y
X
x2-x1
y2-y1
The derivative
x2-x1
y2-y1
xy
xxyy
slope
12
12
xxfxxf
xxxfxf
slope
)()()()( 11
12
12
x1 x2
y1
y2
xxfxxf
slope x
)()(lim 0
xxfxxf
yxfdxdy
x
)()(lim)(' 0
The derivative describes the change in the slope of functions
Aryabhata (476-550)
Bhaskara II (1114-1185)
The first Indian satellite
-10
-5
0
5
10
-4 -2 0 2 4
YX
)()(
)( ufbudxxdf
bauuy
u
0)2(
2
2)2(
bdxxdf
bay
( * ) ' '* * 'f g f g f g ( ( )) ' '* 'f g f g
( ) ' ' '
( ) ' ' '
f g f g
f g f g
'
2
'* * 'f f g f g
g g
Four basic rules to calculate derivatives
b
Local minimum
0)(
dxxdf Stationary point,
point of equilibrium)('
)()(cf
abafbf
Mean value theorem
0
1
2
3
0 5 10 15 20
Y
X
05
10152025303540
0 5 10 15 20
Y
X
Dy=30-10
Dx=15-5
25101030
lim 0
xy
dxdy
x
The derivative of a linear function y=ax equals its slope a
xy 2
Dy=0
0lim 0 xy
dxdy
x
The derivative of a constant y=b is always zero. A constant doesn’t change.
2y
aybaxy '
0
5
10
15
20
25
30
0 1 2 3 4
Y
X
xx edxdy
ey
dy
dx
xeydxdy
The importance of e
ax
x exa
1lim
ex
x
x
11lim
)ln(xy
xedxdy
edydx
dydxdxdy
exxy
yy
y
11
/1
)ln(
-2
-1
0
1
2
3
4
0 1 2 3 4
Y
X
)ln(xy xy
1
1)ln()ln()ln()ln(
)ln()ln(
)1
)ln(00())'ln()(ln(''
bbxbaxbau
uxbab
abxxb
axxbxexbaeuey
eeaxy
baxy
xxxxu
ubxax
bbababxbabbxaabuey
eeaby
)ln()ln()0)ln(10())'ln()(ln(''
)ln()ln(
xaby
xx
xx
xxxxxy
xxxx
y
xx
x
)sin(lim)cos(
)sin()sin()cos()cos()sin(lim'
)sin()sin(lim'
00
0
)sin(xy
)cos()(sin'1)sin(
lim 0 xxxx
x
)sin()(cos' xx
The approximation of a small increase
xxfxfxxfx
xxfxfxxfx
xfxxfslope
xx
x
x
)('lim)()(lim
0)(')()(
lim
)()(lim
00
0
0
How much larger is a ball of 100 cm radius if we extend its radius to 105 cm?
3233 628.0)1(405.0 mmmV
The true value is DV = 0.66m3.
23 4'34
rVrV
012345678
-4 -2 0 2 4
Y
X
xee
yxx
x
0lim
Rule of l’Hospital
2111
)0(1
'
)(;)(lim 0
yee
y
xxgeexfxee
y
xx
xxxx
x
)()()(
)(
)()()(
)(
0
0
xxdxxdg
dxdxxdg
xg
xxdxxdf
cxdxxdf
xf
)()(
)( 00 xxdxxdf
xf
The value of a function at a point x can be approximated by its tangent at x.
)(')('
)()(
)()(
)()(
0
0
xgxf
cxdxxdg
cxdxxdf
xgxf
)(')('
lim)()(
limxgxf
xgxf
axax
0)(lim)(lim00
xgxf xxxx
0
5
10
15
20
25
30
-2 -1 0 1 2 3 4 5
Y
X
-25
-20
-15
-10
-5
0
5
-2 -1 0 1 2 3 4 5
Y
X
-20-15-10-505
101520
-2 -1 0 1 2 3 4 5Y
X
Stationary points
Minimum MaximumHow to find minima and maxima of functions?
0
5
10
15
20
25
30
-2 -1 0 1 2 3 4 5
Y
X
f’<0f’>0 f’<0
f’=0
f’=0
f(x)
f’(x)
f’’(x)
86''
283'
10242
23
xy
xxy
xxxy
387.2;279.0910
34
2830' 212,12 xxxxxy
Populations of bacteria can sometimes be modelled by a general trigonometric function:
dcbtaN )sin(
-2-10123456
0 2 4 6
Y
X
2)64sin(3 tN
a: amplitudeb: wavelength; 1/b: frequencyc: shift on x-axisd: shift on y-axis
a
b
d
c
23
8264
0)64cos(0)64cos(12
ktkt
ttdtdN
The time series of population growth of a bacterium is modelled by
2)64sin(3 tN At what times t does this population have maximum sizes?
0
5
10
15
20
25
30
-2 -1 0 1 2 3 4 5
Y
X
Maximum and minimum change
Point of maximum changePoint of inflection
f’=0
f’=0
Positive sense
Negative sense
At the point of inflection the first derivative has a maximum or minimum.To find the point of inflection the second derivative has to be zero.
34
860''
86''
283'
10242
23
xxy
xy
xxy
xxxy
4/3
The most important growth process is the logistic growth (Pearl Verhulst model)