1. Notes for Analysis Et/Wi
Second Quarter GS
TU Delft
2001
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2. Week 1. Defining Sequences
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3. Week 1. Convergence of a Sequence
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4. Week 1. Showing Convergence by the Definition
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5. Week 1. Rules for computing the limit
Squeeze Theorem
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6. Week 1. Existence of a limit without computation
Definitions
Increasing Sequence Theorem
And a similar result with `decreasing’,`bounded below’ ….
Bounded = bounded above + bounded below.
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7. Week 1. Example: ‘first we show the limit exists, then we can compute it’
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8. Week 1. Defining a Series
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9. Week 1. Sequences and Series
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10. Week 1. Famous Series I
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11. Week 1. Famous Series II 
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12. Week 2. Comparison of Series by Integrals I
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13. Week 2. Comparison of Series by Integrals II
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14. Week 2. Comparison of Series by Series, directly
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15. Week 2. Comparison of Series by Series, via a Limit
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16. Week 2. Absolutely and Conditionally Convergent I
Theorem
Definitions
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17. Week 2. The Ratio Test
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18. Week 2. The Root Test
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19. Week 2. Convergence and Rearrangement of Sequences
Theorem
Theorem 
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20. Week 3. An Example
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21. Week 3. Power Series
Definition
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22. Week 3. Radius of Convergence
Theorem
No statement if
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23. Week 3. Differentiation of Power Series
Theorem
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24. Week 3. Differentiation of Power Series, `what to prove’
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25. Week 3. Elementary Power Series I
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26. Week 3. Elementary Power Series II
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27. Week 4. Taylor series I, from coefficients to derivatives
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28. Week 4. Taylor series II, from derivatives to coefficients
Taylor series for a = 0 are often called Maclaurin series.
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29. Week 4. Taylor polynomials and Remainders
Taylor’s inequality
This inequality gives an estimate for the difference between the function and the `approximating’ n-th Taylor polynomial.
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30. 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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31. Week 4. The Taylor series at 0 for sin(x)
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32. 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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33. Week 5. Functions from to
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34. Another example, click to rotate
Week 5. Functions from to
Another example, click to rotate
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35. Week 5. Limits of vector functions
Definition
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36. Week 5. The derivative of a vector function
Theorem (rules for differentiation)
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37. Week 5. The derivative and the tangent vector
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38. Week 5. Smooth curve
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39. Week 5. Defining the length of a curve
Definition
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40. 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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41. Week 5. Lengths, curves and polygons II
Notice that here the large || denote lengths, the small || absolute values.
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42. Week 5. Lengths, curves and polygons III
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43. Week 6. Functions of several variables
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44. Week 6. Level curves and contour maps
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45. Week 6. Definition of Limit
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46. Week 6. How to show that the limit exists?
Recipe
Example: limit exists
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47. 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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48. Week 6. Example where the limit does not exist
Example: limit does not exist
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49. Week 6. A not so nice example, I
A non-conclusive picture …. .
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50. Week 6. A not so nice example, II
This parametric plot suggests that the limit does not exist.
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51. 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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52. Week 6. Continuous function.
Definition
Definition
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53. Week 6. Partial derivative.
Definition
Definition
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54. 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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55. Week 6. Partial Derivative and Tangent Line
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56. Week 6. Computing Partial Derivatives
Clairaut’s Theorem (?)
For the partial derivative with respect to x one differentiates with respect to x as if y is a constant (and vice versa). 
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