4A Rate of Change
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Transcript of 4A Rate of Change
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4A Rate of Change
A rate is a quant ity mea sured w ith re sp ect to another mea sured quant ity:a rate of speed of 60 kilometres an hour In general, dy
dx gives the rate of change in y with re sp ect to x .
Example:
a) Typing rate =word sminute
b) Car speed =km
hour
30
60 or km per hour
Average Rate of Change
Average sp eed = S( t 2) S( t 1)t 2 t 1= Slope of the l ine segment
S (distance in km)
t (t ime in hour)
300
500
0 3 7
m1
= 100
m2= 50
KL
Ipoh
Penang
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4A Rate of Change
Average Rate of Change from cur ve gra ph
Average rate(sp eed)
=S( t 2) S( t 1)
t 2 t 1= Slope of the chord
S (distance in km)
t (t ime in hour)
S( t 1)
S( t 2)
0 t 1 t 2
The average rate of change between two point s on the gra phis the slope of the chord connect ing the se two point s.
Chord
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4A Rate of Change
Instantaneou s Rate of Change
The instantaneou s rate(sp eed)
=S (t 1)=
Slope of the tangent
S (distance in km)
t (t ime in hour)0 t 1
The ins ta n ta n eou s rate of change (sp eed in th is ca se) at a part icularpoint is given by the slope of the tangent to the gra ph.
tangent
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4A Rate of Change
dydx
If dy
dx
= Slope of the tangent = Rate of change
> 0, y is in c re a s in g
If dydx
< 0, y is d e c re a s in g
y
x
y
x
T
T
(Instantaneou s rate of change)
As x increa ses, y al so increa ses
As x increa ses, y decrea ses
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4A Rate of ChangeEx
ample:If P re pre sent the price of petrol,where t is the number of year s since 2000
P(t)= 0.1t + 1.22 R ingg it P (t)= 0.1 R ingg it per year
It show s that the price of petrol increa ses by 10 cent s per year
P(t)
t
P(t)= 0.1t + 1.221.22
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Refer to Exe 4A Q2 :
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Q2 (b) Method IY1=20 9/(X+ 5)
Homescreen:
Y-VARS 1:Function 1:Y1Y1(4) = 19t = 4, H = 19 mY1(8) = 19.3 07 6923 1
t = 8, H = 19.3 mY1(12) = 19.4 7 058824t = 12, H = 19.5 m
Y =
VARS
2ND ENTRY
2ND ENTRY
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Q2 (b) Method IIY1=20 9/(X+ 5)
TblStart = 4^ Tbl = 4
t = 4, H = 19 mt = 8, H = 19.3 mt = 12, H = 19.5 m
2nd TBLSET
Y =
2nd TABLE
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Q2 (c) Method I Homescreen
8: nDeriv(
Y-VARS1
:Function1
:Y1
n Deriv (Y 1, X, 0) = 0.36 000001 4t = 0, dH/dt = 0.36 m/yr n Deriv (Y 1, X, 5) = 0.090000001
t = 5, dH/dt = 0.09 m/yr n Deriv (Y 1, X, 10) = 0.0400000005t = 10, dH/dt = 0.04 m/yr
VARS
MATH
2ND ENTRY
2ND ENTRY
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Q2 (c) Method IIY2 = 8: nDeriv(Y2 = n Deriv (Y 1, X, X)
TblStart = 0^ Tbl = 5
t = 0, dH/dt = 0.36 m/yr t = 5, dH/dt = 0.09 m/yr t = 10, dH/dt = 0.04 m/yr
2nd
TBLSET
Y =
2nd TABLE
MATH
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4A Rate of ChangeEx
ample:Exe 4A Q1.Q= 100 10 t where t is the age of a persondQdt =
10 ( )(t)12
12
= 5 (t)12
= t5
b)
(i) t = 25 year s, dQdt =
t
5=
25
5
= 55
= 1
Graphic Calculator:Find dy
dx =Rate of change
U nit per year
2nd CALC
5:X= 25 ENTER
dydx
dydx
= ? -1X= 25
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4A Rate of ChangeD i
rec
tly Proportio
nal
If y is directly proport ional to xy x
y = c x
E
I n ver s ely Proport io n alIf y is inver sely proport ional to x
y x
y = c
E
, where c is a con stant
1
x1
, where c is a con stant
