Notes for Analysis Et/Wi

1
Notes for Analysis Et/Wi

• Second Quarter
• GS
• TU Delft
• 2001

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Week 1. Defining Sequences
3
Week 1. Convergence of a Sequence
4
Week 1. Showing Convergence by the Definition
5
Week 1. Rules for computing the limit
Squeeze Theorem
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Week 1. Existence of a limitwithout computation
Definitions
Increasing Sequence Theorem

• And a similar result with decreasing,bounded
  below .
• Bounded bounded above bounded below.

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Week 1. Examplefirst we show the limit exists,
then we can compute it
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Week 1. Defining a Series
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Week 1. Sequences and Series
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Week 1. Famous Series I
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Week 1. Famous Series II
?
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Week 2. Comparison of Series by Integrals I
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Week 2. Comparison of Series by Integrals II
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Week 2. Comparison of Series by Series, directly
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Week 2. Comparison of Series by Series, via a
Limit
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Week 2. Absolutely and Conditionally Convergent I
Theorem
Definitions
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Week 2. The Ratio Test
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Week 2. The Root Test
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Week 2. Convergence and Rearrangement of Sequences
Theorem
Theorem
?
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Week 3. An Example
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Week 3. Power Series
Definition
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Week 3. Radius of Convergence
Theorem
No statement if .
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Week 3. Differentiation of Power Series
Theorem
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Week 3. Differentiation of Power Series, what
to prove
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Week 3. Elementary Power Series I
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Week 3. Elementary Power Series II
?
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Week 4. Taylor series I,from coefficients to
derivatives
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Week 4. Taylor series II,from derivatives to
coefficients
Taylor series for a 0 are often called
Maclaurin series.
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Week 4. Taylor polynomials and Remainders
Taylors inequality
This inequality gives an estimate for the
difference between the function and the
approximating n-th Taylor polynomial.
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Week 4. When does the Taylor series converge?
Theorem
It is more useful to see how good the polynomial
Tn approximates f.
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Week 4. The Taylor series at 0 for sin(x)
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Week 4. The Binomial Series
For k a non-negative integer the series has only
finitely many tems. For all other k the radius of
convergence is 1.
?
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Week 5. Functions from to 3.
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Week 5. Functions from to 3.
Another example, click to rotate
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Week 5. Limits of vector functions
Definition
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Week 5. The derivativeof a vector function
Theorem (rules for differentiation)
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Week 5. The derivative andthe tangent vector
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Week 5. Smooth curve
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Week 5. Defining the length of a curve
Definition
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Week 5. Lengths, curves and polygons I
The length of a polygon and the arc length of a
curve have been defined in two different ways. In
order to make any sense there should be a
relation between these two definitions. The next
pages show that the arc length of a curve with
bounded second derivatives can be approximated by
the length of nearby polygons.
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Week 5. Lengths, curves and polygons II
Notice that here the large denote lengths, the
small absolute values.
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Week 5. Lengths, curves and polygons III
?
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Week 6. Functions of several variables
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Week 6. Level curves and contour maps
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Week 6. Definition of Limit
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Week 6. How to show that the limit exists?
Recipe
Example limit exists
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Week 6. How to show that the limit does not
exist?
Recipe
In general, before using one of these two
recipes, one has to convince oneself which is the
most likely case to hold!
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Week 6. Example where the limit does not exist
Example limit does not exist
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Week 6. A not so nice example, I
A non-conclusive picture . .
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Week 6. A not so nice example, II
This parametric plot suggests that the limit does
not exist.
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Week 6. A not so nice example, III
Even having the same limit behaviour following
all straight lines is not sufficient for the
limit to exist.
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Week 6. Continuous function.
Definition
Definition
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Week 6. Partial derivative.
Definition
Definition
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Week 6. Partial derivative.
Alternative Definition
Alternative Definition
From this definition one sees which is the best
possible linear approximation in the y-direction.
From this definition one sees which is the best
possible linear approximation in the x-direction.
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Week 6. Partial Derivative and Tangent Line
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Week 6. Computing Partial Derivatives
Clairauts Theorem (?)
For the partial derivative with respect to x one
differentiates with respect to x as if y is a
constant (and vice versa).
?