Calculus 4.5

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4.5 Linear Approximations, Differentials
and Newtons Method
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For any function f (x), the tangent is a close
approximation of the function for some small
distance from the tangent point.
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Start with the point/slope equation
linearization of f at a
The linearization is the equation of the tangent
line, and you can use the old formulas if you
like.
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Linearization Example
Find the linear approximation of f(x) x2 at x
1. Use the approximation to 1.12.
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Linearization Example
Find the linear approximation of f(x) x2 at x
1. Use the approximation to 1.12.
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Linearization Example
Use linearization to approximate
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Linearization Example
Use linearization to approximate
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Approximating Binomial Powers
Approximating Binomial Powers this is a special
case of a general linearization formula that
applies to the powers of 1x for small values of
x
Approximating Binomial Powers
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Approximating Binomial Powers
Use this formula to find polynomials that will
approximate the following functions for values of
x close to zero
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Important linearizations for x near zero
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.
dy can be considered a very small change in y.
dx can be considered a very small change in x.
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Let be a differentiable
function. The differential is an
independent variable. The differential is
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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?
very small change in r
very small change in A
(approximate change in area)
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(approximate change in area)
Compare to actual change
New area
Old area
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Newtons Method
Newtons Method numerical method of
approximating the zeros of a function.
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Newtons Method
Finding a root for
We will use Newtons Method to find the root
between 2 and 3.
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Guess
(not drawn to scale)
(new guess)
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Guess
(new guess)
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Guess
(new guess)
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Guess
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.
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Guess
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.
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Find where crosses .
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There are some limitations to Newtons method
Looking for this root.
Bad guess.
Wrong root found
Failure to converge
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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 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.
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Now lets do one on the TI-89
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Now lets do one on the TI-89
Approximate the positive root of
Select and press
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Calculus Tools
Press (Deriv)
Enter the equation. (You will have to unlock the
alpha mode.)
Set the initial guess to 1.
Set the iterations to 3.
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