AP CALCULUS AB

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AP CALCULUS AB

• Chapter 4
• Applications of Derivatives
• Section 4.5
• Linearization and Newtons Method

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What youll learn about

• Linear Approximation
• Newtons Method
• Differentials
• Estimating Change with Differentials
• Absolute, Relative, and Percent Change
• Sensitivity to Change
• and why
• Engineering and science depend on approximation
  in most practical applications it is important
  to understand how approximation techniques work.

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Linear Approximation

• Any differentiable curve is Locally Linear
• if you zoom in enough times.
• Do Exploration 1 Appreciating Local Linearity
  (p 233)
• A fancy name for the equation of the tangent line
  at a is
• The linearization of
  f at a
• y f(a) f(a)(x a)

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Definition of Linearization
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Just Math TutoringYou Tube

• What is Linearization?
• Just math tutoring
• Finding the Linearization at a point
• Followed by
• 25) Linear Approximation
• 10 minutes total time needed Watch if you miss
  class this day or do not understand!

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Example 1 Finding a Linearization

• Find the linearization of
  at x 0 (center of approximation) and use
  it to approximate without a
  calculator.
• Then use a calculator to determine the accuracy
  of the approximation.
• Point of tangency f
  (0)
• L(x) Equation of the tangent line
• Evaluate L(.02)
• Calculator approximation?
• Approximation error

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You try Find linearization L(x) of f(x) at x a
when and a 2.
How accurate is the approximation L(a 0.1)
f(a 0.1)

• Point of tangency f(2)
  f (2)
• Tangent Line equation L(x)
• Evaluate L(2.1) f(2.1)
• Approximation error

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Example 2 Find the linearization of f(x) cos x
at x p/2 and use it to approximate cos 1.75
without a calculator. Then use a calculator to
determine the accuracy of the approximation.

• Point of tangency f (p/2)
  f (p/2)
• Tangent Line equation
• L(x)
• Evaluate L(1.75) cos 1.75 by calculator
• Approximation error

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Example Finding a Linearization

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Summary

• Every function is locally linear about a point
  x a. If you evaluate the tangent line at x a
  for points close to a, you will have a close
  approximation to the functions actual value.

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Steps

• Using f(x), find the equation of a tangent line
  at some point (a, f(a)).
• Find f(a) by plugging a into f(x).
• Find the slope from f(a).
• L(x) f(a) f(a) (x - a).
• 2) Evaluate L(x) for any x near a to get a
  close approximation of f(x) for points near a.

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Example 3 Approximating Binomial Powers using
the general formula

• Use the formula to find polynomials that will
  approximate the following functions for values of
  x close to zero.
• b)
  c) d)
• How?
• Rewrite expression as (1 x) k,
• Identify coefficients of x and k.
• Find L(x) 1 kx for each expression.

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Example 4 Use linearizations to approximate
roots. Find a) and b)

• Identify function f(x)
• Let a be the perfect square closest to 123. Find
  L(x) at x a.
• Use L(x) to estimate
• Error?
• You try b.

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Differentials

(With dx as in independent variable and dy a
dependent variable that depends on both x and
dx.) Although Liebniz did most of his calculus
using dy and dx as separable entities, he never
quite settled the issue of what they were. To
him, they were infinitesimals nonzero
numbers, but infinitesimally small. There was
much debate about whether such things could exist
in mathematics, but luckily for the early
development of calculus it did not matter thanks
to the Chain Rule, dy/dx behaved like a quotient
whether it was one or not.
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Example Finding the Differential dy

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Example 6 Find the differential dy and evaluate
dy for the given values of x and dx. How?
Find f (x), multiply both sides by dx, evaluate
for given values.

• a) y x5 37x b) y sin 3x c) x y xy
• x1, dx 0.01 xp, dx -0.02
  x2, dx 0.05
• You try

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More Notation
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Example 7 Finding Differentials of functions.
Find dy/dx and multiply both sides by dx.

• d (tan (2x)) b)
• You try d(e5x x5)

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Estimating Change with Differentials

• Suppose we know the value of a differentiable
  function f(x) at a point a and we want to predict
  how much this value will change if we move to a
  nearby point (a dx).
• If dx is small, f and its linearization L at a
  will change by nearly the same amount.
• Since the values of L are simple to calculate,
  calculating the change in L offers a practical
  way to estimate the change in f.

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Differential Estimate of Change

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Estimating Change with Differentials
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Example Estimating Change with Differentials

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Example 8 The radius r of a circle increases
from a 10 to 10.1 m. Use dA to estimate the
increase in the circles area A. Compare this
estimate with the true change ?A, and find the
approximation error.

• Area formula for a circle A
• True change f(10.1) f(10)
• Estimated change dA/dr
• dA
• Approximation error ?A dA
• You try f(x) x3 - x, a 1, dx 0.1

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In Review

• The linear approximation of a differentiable
  function at c is
• because, from the slope of the tangent line

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In Review

• Definition of Differentials
• is a differentiable function in an
  open interval containing x.
• The differential of x is any non-zero
  real number.
• The differential of y is

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Summary

• Linearization The equation of a tangent line to
  f at a point a will give a good approximation of
  the value of a function f at a.
• The Linearization of (1 x)k 1 kx
• Newtons Method is used to find the roots of a
  function by using successive tangent line
  approximations, moving closer and closer to the
  roots of f if you start with a reasonable value
  of a.
• Differentials Differentials simply estimate the
  change in y as it relates to the change in x for
  given values of x. We learned how to estimate
  with linearizations, differentials are simply a
  more efficient method of finding change.

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FYI not testedNewtons Method for
approximating a zero of a function

• Approximate the zero of a function by finding the
  zeros of linearizations converging to an accurate
  approximation.
• Just Math Tutoring Newtons Method
• 
  (729 minutes)

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Procedure for Newtons Method

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Procedure for Newtons Method
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Using Newtons Method