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8/3/2019 Newton-raphson as Calculus
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4.5: Linear Approximations, Differentials
and Newtons Method
Gre Kell Hanford Hi h School Richland Washin ton
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For any functionf(x), the tangent is a close approximationof the function for some small distance from the tangent
point.y
x0 x a!
f x f a!We call the equation of the
tangent the linearization of
the function.
p
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The linearization is the equation of the tangent line, and
you can use the old formulas if you like.
Start with the point/slope equation:
1 1y y m x x ! 1x a! 1y f a! m f ad!
y f a f a x ad !
y f a f a x ad!
L x f a f a x ad! linearization offat a
f x L x} is the standard linear approximation offat a.
p
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Important linearizations forx near zero:
1k
x 1 kx
sin x
cosx
tan x
x
1
x
1
21
1 1 1
2
x x x ! }
1
3 4 4 3
4 4
1 5 1 5
1 51 5 1
3 3
x x
x x
!
} !
f x L x
p
This formula also leads to
non-linear approximations:
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Differentials:
When we first started to talk about derivatives, we said that
becomes when the change in x and change in
y become very small.
y
x
(
(
dy
dx
dy can be considered a very small change in y.
dx can be considered a very small change in x.
p
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Let be a differentiable function.
The differential is an independent variable.
The differential is:
y f x!
dxdy dy f x dxd!
p
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Example: Consider a circle of radius 10. If the radius
increases by 0.1, approximately how much will the area
change?
2A rT!
2dA r dr T!
2dA dr
rdx dx
T!
very small change in A
very small change in r
2 10 0.1dA T!
2dA T! (approximate change in area)
p
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2dA T! (approximate change in area)
Compare to actual change:
New area:
Old area:
2
10.1 102.01T T!
2
10 100.00T T!
2.01T
.01
2.01
T
T
!
Error
Original Area
Error
Actual Answer
.0049751} 0.5%}
0.01%}.0001}.01
100
T
T!
p
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Newtons Method
21
32f x x! Finding a root for:c mputer may nothaveenough memory toopen theimage, or theimagemay havebeen corrupted.Restartyour computer, and then open thefileagain.I fthe red x stillappears, you may havetodelete theimage and then insertit again.
We will use Newtons
Method to find the
root between 2 and 3.
p
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in. Ifthe red x stillappears, you may haveto deletetheimage and then insertit again.
Guess: 3
21
3 3 3 1.52
f ! !
1.5
tangent 3 3m fd! !
21
32
f x x!
f x xd !
z
1.5
1.53
z!
1.5
3z !1.5
3 2.53
!
(not drawn to scale)
(new guess)
p
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in. Ifthe red x stillappears, you may haveto deletetheimage and then insertit again.
Guess: 2.5
21
2.5 2.5 3 .1252
f ! !
1.5
tangent 2.5 2.5m fd! !
21
32
f x x!
f x xd !
z
.125
2.5z !.125
2.5 2.452.5
!
(new guess)
p
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in. Ifthe red x stillappears, you may haveto deletetheimage and then insertit again.
Guess:2.45
2.45 .00125f !
1.5
tangent 2.45 2.45m fd! !
21
32
f x x!
f x xd !
z
.00125
2.45z !
.00125
2.45 2.449489795922.45
! (new guess) p
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Guess: 2.44948979592
2.44948979592 .00000013016f !
Amazingly close to zero!
This is Newtons Method of finding roots. It is an example
of an algorithm (a specific set of computational steps.)
It is sometimes called the Newton-Raphson method
This is a recursive algorithm because a set of steps are
repeated with the previous answer put in the next
repetition. Each repetition is called an iteration.
p
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This is Newtons Method of finding roots. It is an example
of an algorithm (a specific set of computational steps.)
It is sometimes called the Newton-Raphson method
Guess: 2.44948979592
2.44948979592 .00000013016f !
Amazingly close to zero!
Newtons Method:
1
n
n n
n
f xx x
f x !
d
This is a recursive algorithm because a set of steps are
repeated with the previous answer put in the next
repetition. Each repetition is called an iteration.
p
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nx
nf xn
nf xd
1
n
n n
n
f xx x
f x !
d
Find where crosses .3
y x x! 1y !
31 x x!
30 1x x! 3 1f x x x! 23 1f x xd !
0 1 1 21
1 1.5
2
!
1 1.5 .875 5.75.875
1.5 1.34782615.75
!
2 1.3478261 .1006822 4.4499055 1.3252004
3
1.3252004 1.3252004 1.0020584 ! 1}p
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There are some limitations to Newtons method:
Wrong root found
Looking for this root.
Bad guess.
Failure to converge
p
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Newtons method is built in to the Calculus Tools
application on the TI-89.
Of course if you have a TI-89, you could just use
the root finder to answer the problem.
The only reason to use the calculator for Newtons Methodis to help your understanding or to check your work.
It would not be allowed in a college course,
on theAP exam or on one of my tests.
p
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APPS
Select and press .Calculus Tools ENTER
If you see this
screen, press
, change the
mode settings as
necessary, and
press
again.
ENTER
APPS
Now lets do one on the TI-89:
p
3 1f x x x! Approximate the positive root of:
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Now lets do one on the TI-89:
APPS
Select and press .Calculus Tools ENTER
Press (Deriv)F2
Press (Newtons Method)3
Enter the equation.
(You will have to unlockthe alpha mode.)Set the initial guess to 1.
Press .ENTER
3 1f x x x! Approximate the positive root of:
Set the iterations to 3.
p
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Press to see
the summary screen.
ESC
Press to see
each iteration.
ENTER
p
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T
Press and thento return your
calculator to normal.
ESC
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