Python Programming For Mathematics

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Doing Math with Python

Author : Amit Saha
Publisher : No Starch Press
ISBN 13 : 1593277199
Total Pages : 265 pages
Book Rating : 4.5/5 (932 download)

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Book Synopsis Doing Math with Python by : Amit Saha

Download or read book Doing Math with Python written by Amit Saha and published by No Starch Press. This book was released on 2015-08-01 with total page 265 pages. Available in PDF, EPUB and Kindle. Book excerpt: Doing Math with Python shows you how to use Python to delve into high school–level math topics like statistics, geometry, probability, and calculus. You’ll start with simple projects, like a factoring program and a quadratic-equation solver, and then create more complex projects once you’ve gotten the hang of things. Along the way, you’ll discover new ways to explore math and gain valuable programming skills that you’ll use throughout your study of math and computer science. Learn how to: –Describe your data with statistics, and visualize it with line graphs, bar charts, and scatter plots –Explore set theory and probability with programs for coin flips, dicing, and other games of chance –Solve algebra problems using Python’s symbolic math functions –Draw geometric shapes and explore fractals like the Barnsley fern, the Sierpinski triangle, and the Mandelbrot set –Write programs to find derivatives and integrate functions Creative coding challenges and applied examples help you see how you can put your new math and coding skills into practice. You’ll write an inequality solver, plot gravity’s effect on how far a bullet will travel, shuffle a deck of cards, estimate the area of a circle by throwing 100,000 "darts" at a board, explore the relationship between the Fibonacci sequence and the golden ratio, and more. Whether you’re interested in math but have yet to dip into programming or you’re a teacher looking to bring programming into the classroom, you’ll find that Python makes programming easy and practical. Let Python handle the grunt work while you focus on the math. Uses Python 3

Mathematics for the Digital Age and Programming in Python

Author : Maria Litvin
Publisher : Skylight Pub
ISBN 13 : 9780972705585
Total Pages : 320 pages
Book Rating : 4.7/5 (55 download)

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Book Synopsis Mathematics for the Digital Age and Programming in Python by : Maria Litvin

Download or read book Mathematics for the Digital Age and Programming in Python written by Maria Litvin and published by Skylight Pub. This book was released on 2008-01-01 with total page 320 pages. Available in PDF, EPUB and Kindle. Book excerpt: The scripting language, Python, and the mathematics involved in its use.

Math Adventures with Python

Author : Peter Farrell
Publisher : No Starch Press
ISBN 13 : 1593278683
Total Pages : 305 pages
Book Rating : 4.5/5 (932 download)

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Book Synopsis Math Adventures with Python by : Peter Farrell

Download or read book Math Adventures with Python written by Peter Farrell and published by No Starch Press. This book was released on 2019-01-08 with total page 305 pages. Available in PDF, EPUB and Kindle. Book excerpt: Learn math by getting creative with code! Use the Python programming language to transform learning high school-level math topics like algebra, geometry, trigonometry, and calculus! Math Adventures with Python will show you how to harness the power of programming to keep math relevant and fun. With the aid of the Python programming language, you'll learn how to visualize solutions to a range of math problems as you use code to explore key mathematical concepts like algebra, trigonometry, matrices, and cellular automata. Once you've learned the programming basics like loops and variables, you'll write your own programs to solve equations quickly, make cool things like an interactive rainbow grid, and automate tedious tasks like factoring numbers and finding square roots. You'll learn how to write functions to draw and manipulate shapes, create oscillating sine waves, and solve equations graphically. You'll also learn how to: - Draw and transform 2D and 3D graphics with matrices - Make colorful designs like the Mandelbrot and Julia sets with complex numbers - Use recursion to create fractals like the Koch snowflake and the Sierpinski triangle - Generate virtual sheep that graze on grass and multiply autonomously - Crack secret codes using genetic algorithms As you work through the book's numerous examples and increasingly challenging exercises, you'll code your own solutions, create beautiful visualizations, and see just how much more fun math can be!

Math for Programmers

Author : Paul Orland
Publisher : Manning Publications
ISBN 13 : 1617295353
Total Pages : 686 pages
Book Rating : 4.6/5 (172 download)

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Book Synopsis Math for Programmers by : Paul Orland

Download or read book Math for Programmers written by Paul Orland and published by Manning Publications. This book was released on 2021-01-12 with total page 686 pages. Available in PDF, EPUB and Kindle. Book excerpt: In Math for Programmers you’ll explore important mathematical concepts through hands-on coding. Filled with graphics and more than 300 exercises and mini-projects, this book unlocks the door to interesting–and lucrative!–careers in some of today’s hottest fields. As you tackle the basics of linear algebra, calculus, and machine learning, you’ll master the key Python libraries used to turn them into real-world software applications. Summary To score a job in data science, machine learning, computer graphics, and cryptography, you need to bring strong math skills to the party. Math for Programmers teaches the math you need for these hot careers, concentrating on what you need to know as a developer. Filled with lots of helpful graphics and more than 200 exercises and mini-projects, this book unlocks the door to interesting–and lucrative!–careers in some of today’s hottest programming fields. Purchase of the print book includes a free eBook in PDF, Kindle, and ePub formats from Manning Publications. About the technology Skip the mathematical jargon: This one-of-a-kind book uses Python to teach the math you need to build games, simulations, 3D graphics, and machine learning algorithms. Discover how algebra and calculus come alive when you see them in code! About the book In Math for Programmers you’ll explore important mathematical concepts through hands-on coding. Filled with graphics and more than 300 exercises and mini-projects, this book unlocks the door to interesting–and lucrative!–careers in some of today’s hottest fields. As you tackle the basics of linear algebra, calculus, and machine learning, you’ll master the key Python libraries used to turn them into real-world software applications. What's inside Vector geometry for computer graphics Matrices and linear transformations Core concepts from calculus Simulation and optimization Image and audio processing Machine learning algorithms for regression and classification About the reader For programmers with basic skills in algebra. About the author Paul Orland is a programmer, software entrepreneur, and math enthusiast. He is co-founder of Tachyus, a start-up building predictive analytics software for the energy industry. You can find him online at www.paulor.land. Table of Contents 1 Learning math with code PART I - VECTORS AND GRAPHICS 2 Drawing with 2D vectors 3 Ascending to the 3D world 4 Transforming vectors and graphics 5 Computing transformations with matrices 6 Generalizing to higher dimensions 7 Solving systems of linear equations PART 2 - CALCULUS AND PHYSICAL SIMULATION 8 Understanding rates of change 9 Simulating moving objects 10 Working with symbolic expressions 11 Simulating force fields 12 Optimizing a physical system 13 Analyzing sound waves with a Fourier series PART 3 - MACHINE LEARNING APPLICATIONS 14 Fitting functions to data 15 Classifying data with logistic regression 16 Training neural networks

Matematicas Y Programacion En Python

Author : J. C. Bautista
Publisher :
ISBN 13 : 9781291711097
Total Pages : 314 pages
Book Rating : 4.7/5 (11 download)

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Book Synopsis Matematicas Y Programacion En Python by : J. C. Bautista

Download or read book Matematicas Y Programacion En Python written by J. C. Bautista and published by . This book was released on 2014-04 with total page 314 pages. Available in PDF, EPUB and Kindle. Book excerpt: Hemos elaborado 120 programas de Python y mas de 110 ilustraciones en una obra que sera util tanto a estudiantes de ciencias de los primeros cursos universitarios, como a estudiantes y profesores de bachillerato.Puedes descargar gratuitamente un extracto de 78 paginas del libro en pdf desde nuestro sitio web www.pysamples.com. Los compradores del libro pueden descargar gratuitamente el codigo de los programas. Todos los programas y graficas se han elaborado utilizando software libre, que se puede obtener gratuita y legalmente en internet, por lo que no es necesario adquirir ningun software. Se puede programar gratuitamente en Python tanto si eres usuario de Windows como si prefieres Mac o Linux. El libro puede ser utilizado como libro de matematicas con el que tambien se puede aprender a programar, o bien como un libro de programacion en Python orientada a las ciencias.

Applying Math with Python

Author : Sam Morley
Publisher : Packt Publishing Ltd
ISBN 13 : 1838986561
Total Pages : 353 pages
Book Rating : 4.8/5 (389 download)

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Book Synopsis Applying Math with Python by : Sam Morley

Download or read book Applying Math with Python written by Sam Morley and published by Packt Publishing Ltd. This book was released on 2020-07-31 with total page 353 pages. Available in PDF, EPUB and Kindle. Book excerpt: Discover easy-to-follow solutions and techniques to help you to implement applied mathematical concepts such as probability, calculus, and equations using Python's numeric and scientific libraries Key FeaturesCompute complex mathematical problems using programming logic with the help of step-by-step recipesLearn how to utilize Python's libraries for computation, mathematical modeling, and statisticsDiscover simple yet effective techniques for solving mathematical equations and apply them in real-world statisticsBook Description Python, one of the world's most popular programming languages, has a number of powerful packages to help you tackle complex mathematical problems in a simple and efficient way. These core capabilities help programmers pave the way for building exciting applications in various domains, such as machine learning and data science, using knowledge in the computational mathematics domain. The book teaches you how to solve problems faced in a wide variety of mathematical fields, including calculus, probability, statistics and data science, graph theory, optimization, and geometry. You'll start by developing core skills and learning about packages covered in Python's scientific stack, including NumPy, SciPy, and Matplotlib. As you advance, you'll get to grips with more advanced topics of calculus, probability, and networks (graph theory). After you gain a solid understanding of these topics, you'll discover Python's applications in data science and statistics, forecasting, geometry, and optimization. The final chapters will take you through a collection of miscellaneous problems, including working with specific data formats and accelerating code. By the end of this book, you'll have an arsenal of practical coding solutions that can be used and modified to solve a wide range of practical problems in computational mathematics and data science. What you will learnGet familiar with basic packages, tools, and libraries in Python for solving mathematical problemsExplore various techniques that will help you to solve computational mathematical problemsUnderstand the core concepts of applied mathematics and how you can apply them in computer scienceDiscover how to choose the most suitable package, tool, or technique to solve a certain problemImplement basic mathematical plotting, change plot styles, and add labels to the plots using MatplotlibGet to grips with probability theory with the Bayesian inference and Markov Chain Monte Carlo (MCMC) methodsWho this book is for This book is for professional programmers and students looking to solve mathematical problems computationally using Python. Advanced mathematics knowledge is not a requirement, but a basic knowledge of mathematics will help you to get the most out of this book. The book assumes familiarity with Python concepts of data structures.

Mathematical Logic through Python

Author : Yannai A. Gonczarowski
Publisher : Cambridge University Press
ISBN 13 : 1108957692
Total Pages : 286 pages
Book Rating : 4.1/5 (89 download)

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Book Synopsis Mathematical Logic through Python by : Yannai A. Gonczarowski

Download or read book Mathematical Logic through Python written by Yannai A. Gonczarowski and published by Cambridge University Press. This book was released on 2022-07-31 with total page 286 pages. Available in PDF, EPUB and Kindle. Book excerpt: Using a unique pedagogical approach, this text introduces mathematical logic by guiding students in implementing the underlying logical concepts and mathematical proofs via Python programming. This approach, tailored to the unique intuitions and strengths of the ever-growing population of programming-savvy students, brings mathematical logic into the comfort zone of these students and provides clarity that can only be achieved by a deep hands-on understanding and the satisfaction of having created working code. While the approach is unique, the text follows the same set of topics typically covered in a one-semester undergraduate course, including propositional logic and first-order predicate logic, culminating in a proof of Gödel's completeness theorem. A sneak peek to Gödel's incompleteness theorem is also provided. The textbook is accompanied by an extensive collection of programming tasks, code skeletons, and unit tests. Familiarity with proofs and basic proficiency in Python is assumed.

A Primer on Scientific Programming with Python

Author : Hans Petter Langtangen
Publisher : Springer
ISBN 13 : 3662498871
Total Pages : 942 pages
Book Rating : 4.6/5 (624 download)

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Book Synopsis A Primer on Scientific Programming with Python by : Hans Petter Langtangen

Download or read book A Primer on Scientific Programming with Python written by Hans Petter Langtangen and published by Springer. This book was released on 2016-07-28 with total page 942 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book serves as a first introduction to computer programming of scientific applications, using the high-level Python language. The exposition is example and problem-oriented, where the applications are taken from mathematics, numerical calculus, statistics, physics, biology and finance. The book teaches "Matlab-style" and procedural programming as well as object-oriented programming. High school mathematics is a required background and it is advantageous to study classical and numerical one-variable calculus in parallel with reading this book. Besides learning how to program computers, the reader will also learn how to solve mathematical problems, arising in various branches of science and engineering, with the aid of numerical methods and programming. By blending programming, mathematics and scientific applications, the book lays a solid foundation for practicing computational science. From the reviews: Langtangen ... does an excellent job of introducing programming as a set of skills in problem solving. He guides the reader into thinking properly about producing program logic and data structures for modeling real-world problems using objects and functions and embracing the object-oriented paradigm. ... Summing Up: Highly recommended. F. H. Wild III, Choice, Vol. 47 (8), April 2010 Those of us who have learned scientific programming in Python ‘on the streets’ could be a little jealous of students who have the opportunity to take a course out of Langtangen’s Primer.” John D. Cook, The Mathematical Association of America, September 2011 This book goes through Python in particular, and programming in general, via tasks that scientists will likely perform. It contains valuable information for students new to scientific computing and would be the perfect bridge between an introduction to programming and an advanced course on numerical methods or computational science. Alex Small, IEEE, CiSE Vol. 14 (2), March /April 2012 “This fourth edition is a wonderful, inclusive textbook that covers pretty much everything one needs to know to go from zero to fairly sophisticated scientific programming in Python...” Joan Horvath, Computing Reviews, March 2015

Programming for Computations - Python

Author : Svein Linge
Publisher : Springer
ISBN 13 : 3319324284
Total Pages : 244 pages
Book Rating : 4.3/5 (193 download)

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Book Synopsis Programming for Computations - Python by : Svein Linge

Download or read book Programming for Computations - Python written by Svein Linge and published by Springer. This book was released on 2016-07-25 with total page 244 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents computer programming as a key method for solving mathematical problems. There are two versions of the book, one for MATLAB and one for Python. The book was inspired by the Springer book TCSE 6: A Primer on Scientific Programming with Python (by Langtangen), but the style is more accessible and concise, in keeping with the needs of engineering students. The book outlines the shortest possible path from no previous experience with programming to a set of skills that allows the students to write simple programs for solving common mathematical problems with numerical methods in engineering and science courses. The emphasis is on generic algorithms, clean design of programs, use of functions, and automatic tests for verification.

Introduction to Scientific Programming with Python

Author : Joakim Sundnes
Publisher :
ISBN 13 : 3030503569
Total Pages : 157 pages
Book Rating : 4.0/5 (35 download)

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Book Synopsis Introduction to Scientific Programming with Python by : Joakim Sundnes

Download or read book Introduction to Scientific Programming with Python written by Joakim Sundnes and published by . This book was released on 2020 with total page 157 pages. Available in PDF, EPUB and Kindle. Book excerpt: This open access book offers an initial introduction to programming for scientific and computational applications using the Python programming language. The presentation style is compact and example-based, making it suitable for students and researchers with little or no prior experience in programming. The book uses relevant examples from mathematics and the natural sciences to present programming as a practical toolbox that can quickly enable readers to write their own programs for data processing and mathematical modeling. These tools include file reading, plotting, simple text analysis, and using NumPy for numerical computations, which are fundamental building blocks of all programs in data science and computational science. At the same time, readers are introduced to the fundamental concepts of programming, including variables, functions, loops, classes, and object-oriented programming. Accordingly, the book provides a sound basis for further computer science and programming studies.

Statistical Learning with Math and Python

Author : Joe Suzuki
Publisher : Springer Nature
ISBN 13 : 981157877X
Total Pages : 261 pages
Book Rating : 4.8/5 (115 download)

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Book Synopsis Statistical Learning with Math and Python by : Joe Suzuki

Download or read book Statistical Learning with Math and Python written by Joe Suzuki and published by Springer Nature. This book was released on 2021-08-03 with total page 261 pages. Available in PDF, EPUB and Kindle. Book excerpt: The most crucial ability for machine learning and data science is mathematical logic for grasping their essence rather than knowledge and experience. This textbook approaches the essence of machine learning and data science by considering math problems and building Python programs. As the preliminary part, Chapter 1 provides a concise introduction to linear algebra, which will help novices read further to the following main chapters. Those succeeding chapters present essential topics in statistical learning: linear regression, classification, resampling, information criteria, regularization, nonlinear regression, decision trees, support vector machines, and unsupervised learning. Each chapter mathematically formulates and solves machine learning problems and builds the programs. The body of a chapter is accompanied by proofs and programs in an appendix, with exercises at the end of the chapter. Because the book is carefully organized to provide the solutions to the exercises in each chapter, readers can solve the total of 100 exercises by simply following the contents of each chapter. This textbook is suitable for an undergraduate or graduate course consisting of about 12 lectures. Written in an easy-to-follow and self-contained style, this book will also be perfect material for independent learning.

Math Coding

Author : Farukh Khalilov
Publisher :
ISBN 13 :
Total Pages : 86 pages
Book Rating : 4.6/5 (423 download)

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Book Synopsis Math Coding by : Farukh Khalilov

Download or read book Math Coding written by Farukh Khalilov and published by . This book was released on 2020-05 with total page 86 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is designed for middle school students and new programming language learners. Computer science has continuously escalated in popularity over the last decade, as students are increasingly showing interest in coding at a young age. In this book you will find a total of 150 math questions, ranging in difficulty from beginner to advanced, with accompanying Python programming language solutions. Python is one of the most popular coding languages and is comparatively easy to learn. With this book, students will be able to increase their proficiency in coding and math computing. This book can be used as a reference for math and computer science teachers for interdisciplinary purposes and will help students improve their skills and critical thinking.

Mathematical Programming for Power Systems Operation

Author : Alejandro Garcés
Publisher : John Wiley & Sons
ISBN 13 : 1119747287
Total Pages : 293 pages
Book Rating : 4.1/5 (197 download)

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Book Synopsis Mathematical Programming for Power Systems Operation by : Alejandro Garcés

Download or read book Mathematical Programming for Power Systems Operation written by Alejandro Garcés and published by John Wiley & Sons. This book was released on 2021-12-01 with total page 293 pages. Available in PDF, EPUB and Kindle. Book excerpt: Explore the theoretical foundations and real-world power system applications of convex programming In Mathematical Programming for Power System Operation with Applications in Python, Professor Alejandro Garces delivers a comprehensive overview of power system operations models with a focus on convex optimization models and their implementation in Python. Divided into two parts, the book begins with a theoretical analysis of convex optimization models before moving on to related applications in power systems operations. The author eschews concepts of topology and functional analysis found in more mathematically oriented books in favor of a more natural approach. Using this perspective, he presents recent applications of convex optimization in power system operations problems. Mathematical Programming for Power System Operation with Applications in Python uses Python and CVXPY as tools to solve power system optimization problems and includes models that can be solved with the presented framework. The book also includes: A thorough introduction to power system operation, including economic and environmental dispatch, optimal power flow, and hosting capacity Comprehensive explorations of the mathematical background of power system operation, including quadratic forms and norms and the basic theory of optimization Practical discussions of convex functions and convex sets, including affine and linear spaces, politopes, balls, and ellipsoids In-depth examinations of convex optimization, including global optimums, and first and second order conditions Perfect for undergraduate students with some knowledge in power systems analysis, generation, or distribution, Mathematical Programming for Power System Operation with Applications in Python is also an ideal resource for graduate students and engineers practicing in the area of power system optimization.

Hacking Math Class with Python

Author : Peter A. Farrell
Publisher : Createspace Independent Publishing Platform
ISBN 13 : 9781508656944
Total Pages : 144 pages
Book Rating : 4.6/5 (569 download)

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Book Synopsis Hacking Math Class with Python by : Peter A. Farrell

Download or read book Hacking Math Class with Python written by Peter A. Farrell and published by Createspace Independent Publishing Platform. This book was released on 2015-02-26 with total page 144 pages. Available in PDF, EPUB and Kindle. Book excerpt: A new kind of math book! Explore math topics from arithmetic to calculus by creating your own graphing and solving tools using Python. Create 2D and 3D graphics, harmonograph and spirograph designs, and fractals in this interactive and visual exploration of mathematics. "A great resource to play with Math and Python via the turtle module, solving equations numerically and 3D graphics via Pi3D." - Amit Saha, author of Doing Math With Python Imagine learning math and Python programming at the same time! You'll learn to use loops, variables, functions, conditionals and lists and apply them to all your math problems. No previous computer experience is required.

A Programmer's Introduction to Mathematics

Author : Jeremy Kun
Publisher :
ISBN 13 :
Total Pages : 400 pages
Book Rating : 4.6/5 (253 download)

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Book Synopsis A Programmer's Introduction to Mathematics by : Jeremy Kun

Download or read book A Programmer's Introduction to Mathematics written by Jeremy Kun and published by . This book was released on 2020-05-17 with total page 400 pages. Available in PDF, EPUB and Kindle. Book excerpt: A Programmer's Introduction to Mathematics uses your familiarity with ideas from programming and software to teach mathematics. You'll learn about the central objects and theorems of mathematics, including graphs, calculus, linear algebra, eigenvalues, optimization, and more. You'll also be immersed in the often unspoken cultural attitudes of mathematics, learning both how to read and write proofs while understanding why mathematics is the way it is. Between each technical chapter is an essay describing a different aspect of mathematical culture, and discussions of the insights and meta-insights that constitute mathematical intuition. As you learn, we'll use new mathematical ideas to create wondrous programs, from cryptographic schemes to neural networks to hyperbolic tessellations. Each chapter also contains a set of exercises that have you actively explore mathematical topics on your own. In short, this book will teach you to engage with mathematics. A Programmer's Introduction to Mathematics is written by Jeremy Kun, who has been writing about math and programming for 10 years on his blog "Math Intersect Programming." As of 2020, he works in datacenter optimization at Google.The second edition includes revisions to most chapters, some reorganized content and rewritten proofs, and the addition of three appendices.

Coding in Python and Elements of Discrete Mathematics

Author : Maria Litvin
Publisher :
ISBN 13 : 9780997252842
Total Pages : 420 pages
Book Rating : 4.2/5 (528 download)

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Book Synopsis Coding in Python and Elements of Discrete Mathematics by : Maria Litvin

Download or read book Coding in Python and Elements of Discrete Mathematics written by Maria Litvin and published by . This book was released on 2019-06-15 with total page 420 pages. Available in PDF, EPUB and Kindle. Book excerpt:

The Statistics and Calculus with Python Workshop

Author : Peter Farrell
Publisher : Packt Publishing Ltd
ISBN 13 : 1800208367
Total Pages : 739 pages
Book Rating : 4.8/5 (2 download)

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Book Synopsis The Statistics and Calculus with Python Workshop by : Peter Farrell

Download or read book The Statistics and Calculus with Python Workshop written by Peter Farrell and published by Packt Publishing Ltd. This book was released on 2020-08-18 with total page 739 pages. Available in PDF, EPUB and Kindle. Book excerpt: With examples and activities that help you achieve real results, applying calculus and statistical methods relevant to advanced data science has never been so easy Key FeaturesDiscover how most programmers use the main Python libraries when performing statistics with PythonUse descriptive statistics and visualizations to answer business and scientific questionsSolve complicated calculus problems, such as arc length and solids of revolution using derivatives and integralsBook Description Are you looking to start developing artificial intelligence applications? Do you need a refresher on key mathematical concepts? Full of engaging practical exercises, The Statistics and Calculus with Python Workshop will show you how to apply your understanding of advanced mathematics in the context of Python. The book begins by giving you a high-level overview of the libraries you'll use while performing statistics with Python. As you progress, you'll perform various mathematical tasks using the Python programming language, such as solving algebraic functions with Python starting with basic functions, and then working through transformations and solving equations. Later chapters in the book will cover statistics and calculus concepts and how to use them to solve problems and gain useful insights. Finally, you'll study differential equations with an emphasis on numerical methods and learn about algorithms that directly calculate values of functions. By the end of this book, you'll have learned how to apply essential statistics and calculus concepts to develop robust Python applications that solve business challenges. What you will learnGet to grips with the fundamental mathematical functions in PythonPerform calculations on tabular datasets using pandasUnderstand the differences between polynomials, rational functions, exponential functions, and trigonometric functionsUse algebra techniques for solving systems of equationsSolve real-world problems with probabilitySolve optimization problems with derivatives and integralsWho this book is for If you are a Python programmer who wants to develop intelligent solutions that solve challenging business problems, then this book is for you. To better grasp the concepts explained in this book, you must have a thorough understanding of advanced mathematical concepts, such as Markov chains, Euler's formula, and Runge-Kutta methods as the book only explains how these techniques and concepts can be implemented in Python.