Notes for Analysis Et/Wi Second Quarter GS TU Delft 2001 11/17/2018

Week 1. Defining Sequences 11/17/2018

Week 1. Convergence of a Sequence 11/17/2018

Week 1. Showing Convergence by the Definition 11/17/2018

Week 1. Rules for computing the limit Squeeze Theorem 11/17/2018

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. 11/17/2018

Week 1. Example: ‘first we show the limit exists, then we can compute it’ 11/17/2018

Week 1. Defining a Series 11/17/2018

Week 1. Sequences and Series 11/17/2018

Week 1. Famous Series I 11/17/2018

Week 1. Famous Series II  11/17/2018

Week 2. Comparison of Series by Integrals I 11/17/2018

Week 2. Comparison of Series by Integrals II 11/17/2018

Week 2. Comparison of Series by Series, directly 11/17/2018

Week 2. Comparison of Series by Series, via a Limit 11/17/2018

Week 2. Absolutely and Conditionally Convergent I Theorem Definitions 11/17/2018

Week 2. The Ratio Test 11/17/2018

Week 2. The Root Test 11/17/2018

Week 2. Convergence and Rearrangement of Sequences Theorem Theorem  11/17/2018

Week 3. An Example 11/17/2018

Week 3. Power Series Definition 11/17/2018

Week 3. Radius of Convergence Theorem No statement if . 11/17/2018

Week 3. Differentiation of Power Series Theorem 11/17/2018

Week 3. Differentiation of Power Series, `what to prove’ 11/17/2018

Week 3. Elementary Power Series I 11/17/2018

Week 3. Elementary Power Series II  11/17/2018

Week 4. Taylor series I, from coefficients to derivatives 11/17/2018

Week 4. Taylor series II, from derivatives to coefficients Taylor series for a = 0 are often called Maclaurin series. 11/17/2018

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. 11/17/2018

Week 4. When does the Taylor series converge? Theorem It is more useful to see how good the polynomial Tn approximates f. 11/17/2018

Week 4. The Taylor series at 0 for sin(x) 11/17/2018

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.  11/17/2018

Week 5. Functions from to 3. 11/17/2018

Another example, click to rotate Week 5. Functions from to 3. Another example, click to rotate 11/17/2018

Week 5. Limits of vector functions Definition 11/17/2018

Week 5. The derivative of a vector function Theorem (rules for differentiation) 11/17/2018

Week 5. The derivative and the tangent vector 11/17/2018

Week 5. Smooth curve 11/17/2018

Week 5. Defining the length of a curve Definition 11/17/2018

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. 11/17/2018

Week 5. Lengths, curves and polygons II Notice that here the large || denote lengths, the small || absolute values. 11/17/2018

Week 5. Lengths, curves and polygons III  11/17/2018

Week 6. Functions of several variables 11/17/2018

Week 6. Level curves and contour maps 11/17/2018

Week 6. Definition of Limit 11/17/2018

Week 6. How to show that the limit exists? Recipe Example: limit exists 11/17/2018

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! 11/17/2018

Week 6. Example where the limit does not exist Example: limit does not exist 11/17/2018

Week 6. A not so nice example, I A non-conclusive picture …. . 11/17/2018

Week 6. A not so nice example, II This parametric plot suggests that the limit does not exist. 11/17/2018

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. 11/17/2018

Week 6. Continuous function. Definition Definition 11/17/2018

Week 6. Partial derivative. Definition Definition 11/17/2018

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. 11/17/2018

Week 6. Partial Derivative and Tangent Line 11/17/2018

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).  11/17/2018