Analysis With Mathematica Vol 1 Single Variable Calculus 1st Edition by Galina Filipuk, Andrzej Kozlowski ISBN 9783110590135 3110590131

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Product details:

ISBN 10: 3110590131 
ISBN 13: 9783110590135
Author: Galina Filipuk, Andrzej Kozlowski

Analysis With Mathematica Vol 1 Single Variable Calculus 1st Edition Table of contents:

1 Number systems

1.1 Sets

1.2 Domains

1.3 Assumptions in Mathematica®

1.4 Quantifiers

1.5 Complex numbers

1.6 Real numbers

1.7 Infinities

1.8 Integers and the Principle of Mathematical Induction

1.8.1 Example

1.8.2 Example

1.9 Algebraic equations and algebraic numbers

1.10 Non-algebraic equations

1.11 Sequences of real numbers and their limits

1.11.1 Example

1.11.2 Example: the number e

1.12 Supremum and infimum

1.12.1 Example

2 Recursive sequences, discrete dynamical systems and their limits

2.1 Example

2.2 Example: the Fibonacci sequence

3 Series

3.1 Sequences and series

3.2 The functions Sum and NSum

3.3 Absolute convergence

3.4 Convergence of series with terms of constant signs

3.4.1 Example

3.5 Convergence of series with terms of non-constant signs

3.5.1 Grouping of terms

3.5.2 Example

3.5.3 Abel’s summation formula

3.5.4 Dirichlet’s and Abel’s tests

3.5.5 Example

3.6 The function SumConvergence

3.6.1 Example

3.7 Riemann’s theorem on conditionally convergent series

3.8 The Cauchy product of series

3.9 Divergent series

3.10 Power series

4 Limits of functions and continuity

4.1 Limits of functions

4.2 One-sided limits

4.3 Continuous functions

4.4 Discontinuous functions

4.4.1 Example: the Dirichlet function

4.5 The main theorems on continuous functions

4.5.1 Example

4.6 Inverse functions and their continuity

4.7 Example: recursive sequences and continuity

4.8 Uniform continuity and the Lipschitz property

4.8.1 Example

5 Differentiation

5.1 Difference quotient and derivative of a function

5.2 Differentiation in Mathematica®

5.2.1 Differentiation of expressions using D

5.2.2 Differentiation of functions using Derivative

5.2.3 Algebraic rules of differentiation

5.2.4 Example: user-defined derivative

5.3 Main properties of differentiable functions

5.3.1 Example: global and local extrema

5.3.2 Example

5.3.3 Example: the inverse function of the hyperbolic sine

5.4 Convex functions

5.4.1 Jensen’s inequality

6 Sequences and series of functions

6.1 Power series continued

6.1.1 Example

6.1.2 Example

6.2 Taylor polynomials and Taylor series

6.2.1 Example

6.2.2 Example

6.2.3 Approximating functions by Taylor polynomials

6.2.4 Example: rational approximation of e

6.2.5 Example: illustration of approximation of functions with Taylor polynomials

6.3 Convergence of sequences and series of functions

6.3.1 Examples: pointwise, uniform and almost uniform convergence of function sequences

6.3.2 Continuity and differentiability of limits and sums

6.3.3 Examples: pointwise, uniform and almost uniform convergence of function series

6.3.4 Example

7 Integration

7.1 Indefinite integrals

7.2 The Risch algorithm

7.2.1 Differential algebras

7.2.2 Example 1: integration of rational functions

7.2.3 Example 2: the Risch algorithm for an exponential extension

7.2.4 Limitations of Mathematica®’s integration

7.3 The Riemann integral

7.3.1 Using Integrate and NIntegrate with definite integrals

7.3.2 Riemann sums

7.4 Improper integrals

7.4.1 Integrals over infinite intervals (improper integrals of the first type)

7.4.2 Improper integrals of the first type and infinite sums

7.4.3 Integrals of unbounded functions (improper integrals of the second type)

Bibliography

Subject Index

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