4: the function © christine crisp “teach a level maths” vol. 2: a2 core modules
TRANSCRIPT
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4: The function 4: The function xey
© Christine Crisp
““Teach A Level Maths”Teach A Level Maths”
Vol. 2: A2 Core Vol. 2: A2 Core ModulesModules
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xey
"Certain images and/or photos on this presentation are the copyrighted property of JupiterImages and are being used with permission under license. These images and/or photos may not be copied or downloaded without permission from JupiterImages"
Module C3
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The Gradient of
xay
growth
Functions of this type, with a > 1, are called functions.
xay We’ve already met the functione.g.
xy 2
xy 3xy 4
)1,0(
Autograph demo
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The Gradient of
xay xay We will now investigate the
gradient of Notice first that as x increases, y increases
e.g.
xy 2
),2(41x x
),1(21
x )1,0(x )2,1(
x )4,2(
x )8,3(
Autograph demo
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The Gradient of
xay
dx
dy
dx
dy
xy 2gradient
Notice first that as x increases, y increases. . . and the also increases
e.g.
x x x x
x
x
The gradient function
x x x x x
x
xay We will now investigate the gradient of
looks
It the same asbut . . .
x2
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The Gradient of
xay
xy 2
dx
dy
We will now investigate the gradient of e.g.
x
The gradient function
x 55dx
dy
8ye.g. gradie
nt
Notice first that as x increases, y increases. . . and the also increases
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The Gradient of
xay
xy 2
x
dx
dy)2(69.0
What do you think will happen if we repeat the process for ? xy 3Well, goes up more steeply than so we get a similar result but the gradient function is above the curve.
xy 3 xy 2
Putting the 2 graphs on the same axes . . .
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The Gradient of
xay
It can be shown that
xx
dx
dyy 3)10.1(3
xy 2 xy 3x
dx
dy3)101(
x
dx
dy2)690(
So,
suggesting that there is a value of a between 2 and 3 where the gradient of is equal to .
xay xa
The 1st gradient graph is under the original curve . . .and the 2nd is above the curve . . .
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The Gradient of
xay
gradient of equalsxa
)3(7182 d.p.e
The value of a where the is an irrational number, written as e, where
xay
xay
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The Gradient of
xay
Using a letter for an irrational number isn’t a new idea to you.
xey
gradient of equalsxa
)3(7182 d.p.e
The value of a where the is an irrational number, written as e, where
xay
You have used ( the Greek p ) for )..3(1423 pd
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The Gradient of
xay
xey
gradient of equalsxa
)3(7182 d.p.e
The value of a where the is an irrational number, written as e, where
xay
xx edx
dyey
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xey
More Indices and Logs
xey The function contains the index x, so x is a log.BUT the base of the log is e not 10, so
( since an index is a log )
We write as ( n for natural ) so,
elog ln
yxey x ln
Logs with a base e are called natural logs
We know that yxy x
10log10
yxey ex log
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xey
xey
xy
xey
The Inverse of xey
We can sketch the inverse by reflecting in y = x.
is a one-to-one function so has an inverse function.
xexf )(
Finding the equation of the inverse function is easy!
So, xxf ln)(1
0xN.B. The domain is .
So
xy ln
xy ln
Forwards x e it = f(x)Backwards opposite of e it is ln it
Autograph demo
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xey
SUMMARY xexf )(• is a growth function.
7182e• (3 d.p.)
xey • At every point on , the gradient equals y:
xx edx
dyey
• The inverse of is
xexf )(
xxf ln)(1
( log with base e )is defined for x > 0
onlyxln
xy
xy lnxey
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xey
xey
Can you suggest equations for the unlabelled graphs below?
Both graphs are stretches of .xey
HINT:
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xey
xey
This . . . is a stretch with scale factor 2 parallel to the y-axis.The equation is xey 2
xey 2
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xey
xey
This . . . 21 is a stretch with scale factor
parallel to the x-axis.
The equation is xey 2
xey 2
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xey
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xey
The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied.For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.
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xey
More Indices and Logs
yxey ex log
xey The function contains the index x, so x is a log.BUT the base of the log is e not 10, so
We know that yxy x
10log10 ( since an index is a log )
We write as ( n for natural ) so,
elog ln
yxey x ln
Logs with a base e are called natural logs
![Page 21: 4: The function © Christine Crisp “Teach A Level Maths” Vol. 2: A2 Core Modules](https://reader033.vdocuments.us/reader033/viewer/2022061613/551ac0dc550346856e8b57a6/html5/thumbnails/21.jpg)
xey
SUMMARY xexf )(• is a growth function.
7182e• (3 d.p.)
xey • At every point on , the gradient equals y:
xx edx
dyey
• The inverse of is
xexf )(
xxf ln)(1
( log with base e )
xey xy ln
xy
is defined for x > 0 only
xln