linear approx, differentials, newton s method
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4.5: Linear Approximations, Differentials and Newton’s Method
Greg Kelly, Hanford High School, Richland, Washington
For any function f (x), the tangent is a close approximation of the function for some small distance from the tangent point.
y
x0 x a
f x f aWe call the equation of the tangent the linearization of the function.
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 a
y f a f a x a
y f a f a x a
L x f a f a x a linearization of f at a
f x L x is the standard linear approximation of f at a.
Important linearizations for x near zero:
1k
x 1 kx
sin x
cos x
tan x
x
1
x
1
21
1 1 12
x x x
13 4 4 3
4 4
1 5 1 5
1 51 5 1
3 3
x x
x x
f x L x
This formula also leads to non-linear approximations:
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.
Let be a differentiable function.
The differential is an independent variable.
The differential is:
y f xdxdy dy f x dx
Example: Consider a circle of radius 10. If the radius increases by 0.1, approximately how much will the area change?
2A r
2 dA r dr
2 dA dr
rdx dx
very small change in A
very small change in r
2 10 0.1dA
2dA (approximate change in area)
2dA (approximate change in area)
Compare to actual change:
New area:
Old area:
210.1 102.01
210 100.00
2.01
.01
2.01
Error
Original Area
Error
Actual Answer.0049751 0.5%
0.01%.0001.01
100
Newton’s Method
213
2f x x Finding a root for:
We will use Newton’s Method to find the root between 2 and 3.
Guess: 3
213 3 3 1.5
2f
1.5
tangent 3 3m f
213
2f x x
f x x
z
1.5
1.53
z
1.5
3z 1.5
3 2.53
(not drawn to scale)
(new guess)
Guess: 2.5
212.5 2.5 3 .125
2f
1.5
tangent 2.5 2.5m f
213
2f x x
f x x
z
.125
2.5z .125
2.5 2.452.5
(new guess)
Guess: 2.45
2.45 .00125f
1.5
tangent 2.45 2.45m f
213
2f x x
f x x
z
.00125
2.45z
.001252.45 2.44948979592
2.45 (new guess)
Guess: 2.44948979592
2.44948979592 .00000013016f
Amazingly close to zero!
This is Newton’s 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.
This is Newton’s 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!
Newton’s Method: 1
nn n
n
f xx x
f x
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.
nx nf xn nf x 1
nn n
n
f xx x
f x
Find where crosses .3y x x 1y
31 x x 30 1x x 3 1f x x x 23 1f x x
0 1 1 21
1 1.52
1 1.5 .875 5.75.875
1.5 1.34782615.75
2 1.3478261 .1006822 4.4499055 1.3252004
31.3252004 1.3252004 1.0020584 1
There are some limitations to Newton’s method:
Wrong root found
Looking for this root.
Bad guess.
Failure to converge
Newton’s 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 Newton’s Method is to help your understanding or to check your work.
It would not be allowed in a college course, on the AP exam or on one of my tests.
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 let’s do one on the TI-89:
3 1f x x x Approximate the positive root of:
Now let’s do one on the TI-89:
APPS
Select and press . Calculus Tools ENTER
Press (Deriv)F2
Press (Newton’s Method)3
Enter the equation.(You will have to unlock the 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.
Press to see
the summary screen.
ESC
Press to see each iteration.
ENTER
Press and then
to return your
calculator to normal.
ESC
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