This site is devoted to mathematics and its applications. Created and run by Peter Saveliev.

# Calculus Illustrated

### From Mathematics Is A Science

*Calculus Illustrated* by Peter Saveliev

This is an early draft, work in progress. If you have any questions, please email me or use the facebook page. The lectures are on this YouTube channel.

## Contents

## 1 Part R: Review of functions

- Chapter 1. Functions

1 What are relations and functions? 2 The coordinate system for dimension 1 3 The coordinate system for dimension 2 4 How relations and functions emerge... 5 Implicit relations and curves 6 Functions: making relations explicit 7 The graph of a function 8 Elementary functions 9 Monotonicity 10 Polynomials 11 Rational functions 12 Sequences

- Chapter 2. Operations on functions

1 Algebra of functions 2 Functions are transformations of the line 3 Transformations of the axes produce new functions 4 Compositions 5 Inverses

- Chapter 3. Classes of functions

1 Algebraic functions 2 Trigonometric functions 3 Concavity 4 Boundedness 5 Symmetries 6 The exponent 7 The logarithm 8 Change of variables 9 Functions of functions 10 History of functions

## 2 Part I: Differential calculus

- Chapter 4. Sequences and their limits

1 Limits of sequences: large scale trends 2 The definition of limit 3 Algebraic properties of limits of sequences 4 Computing limits 5 More properties of limits of sequences 6 Useful theorems* 7 Famous limits 8 The exponential function and the logarithm

- Chapter 5. Limits and continuity

1 Limits of functions: small scale trends 2 Limits under algebraic operations 3 Discontinuity: what to avoid 4 Continuity under algebraic operations 5 Limits and continuity under compositions 6 Continuity of the inverse 7 More on limits and continuity 8 Global properties of continuous functions 9 Large-scale behavior and asymptotes 10 Limits and infinity 11 Continuity and accuracy 12 The ε-δ definition of limit 13 Flowchart for limit computation

- Chapter 6. The derivative

1 The Tangent Problem 2 The average velocity vs. the instantaneous velocity 3 Instantaneous rate of change: the derivative 4 Derivative as a limit 5 Differentiability 6 Derivative as a function 7 Computing derivatives from the definition 8 Location - velocity - acceleration 9 A ball is thrown...

- Chapter 7. Differentiation

1 Derivatives without limits 2 Derivatives under addition and constant multiple 3 Derivatives under multiplication and division 4 Examples of differentiation 5 Repeated differentiation 6 Derivatives under compositions: the Chain Rule 7 Change of variables and the derivative 8 Implicit differentiation and related rates 9 Radar gun: the math 10 The derivative of the inverse 11 Reversing differentiation: antiderivatives

- Chapter 8. The main theorems of differential calculus

1 Extreme points and the derivative 2 Maximum and minimum values of functions 3 What the derivative says about the average rate of change 4 Monotonicity and the sign of the derivative 5 Concavity and the sign of the second derivative 6 Derivatives and extrema 7 Antiderivatives 8 Using differentiation to compute limits: L'Hopital's Rule

- Chapter 9. Applications of differential calculus

1 Optimization 2 Solving equations numerically: bisection and Newton's method 3 Linearization 4 The accuracy of the best linear approximation 5 Numerical differentiation 6 Flows: a discrete model 7 Motion under forces: a discrete model 8 Exponential models: discrete and continuous 9 Functions of several variables

## 3 Part II: Integral calculus

- Chapter 10. The integral

1 The Area Problem 2 The displacement 3 How to approximate the area under a graph 4 Riemann sums: the setup 5 Riemann sums vs. difference quotients 6 Properties of Riemann sums 7 The sigma notation 8 The Riemann integral 9 Properties of Riemann integrals 10 The Fundamental Theorem of Calculus 11 Integrability*

- Chapter 11. Integration

1 Linear change of variables in integral 2 Integration by substitution: compositions 3 Change of variables in integrals 4 Change of variables in definite integrals 5 Trigonometric substitutions 6 Integration by parts: products 7 Integration methods 8 New functions via integration 9 The areas of infinite regions: improper integrals 10 Properties of proper and improper integrals

- Chapter 12. Applications of integral calculus

1 The coordinate system for dimension 3 2 The area between two graphs 3 Volumes via cross-sections 4 The linear density and the mass 5 Center of mass 6 Volumes of solids of revolution 7 The radial density and the mass 8 Flow rate 9 Work 10 The average value of a function 11 Numerical integration 12 Lengths of curves

- Chapter 13. Series

1 From linear to quadratic approximations 2 Taylor polynomials 3 Sequences of functions 4 Infinite series 5 Examples of series 6 Comparison of series 7 Algebraic properties of series 8 Divergence 9 Series with non-negative terms 10 Comparison of series, continued 11 Absolute convergence 12 The Ratio Test and the Root Test 13 Power series 14 Calculus of power series

- Chapter 14. Several variables

1 A ball is thrown... 2 Introduction to parametric curves 3 Introduction to functions of several variables 4 Introduction to calculus of several variables 5 Differential equations 6 The centroid of a flat object 7 Differential forms 8 Discrete forms*

## 4 Part III: Calculus in higher dimensions

- Chapter 15. Functions in multidimensional spaces

1 Multiple variables, multiple dimensions 2 Euclidean spaces and Cartesian systems of dimensions 1, 2, 3,... 3 Geometry of distances 4 Sequences in ${\bf R}^n$ 5 Vectors 6 Algebra of vectors 7 Components of vectors 8 Lengths of vectors 9 Representing parametric curves with vectors 10 The cycloid 11 Angles between vectors and the dot product 12 Projections

- Chapter 16. Parametric curves

1 Parametric curves 2 Planetary motion 3 Limits of parametric curves 4 Continuity 5 Coordinate-wise treatment 6 The derivative 7 The Riemann sums 8 The Riemann integral 9 Computing derivatives 10 Derivatives of derivatives 11 Reversing differentiation: antiderivatives 12 The speed 13 The arc-length parametrization 14 Lengths of curves 15 The curvature 16 The helix 17 The torsion

- Chapter 17. Functions of several variables

1 Overview of functions 2 Linear functions and planes in R3 3 An example of a non-linear function 4 Graphs 5 Difference quotients 6 Limits 7 Continuity 8 Differentiability and linear approximations 9 Partial differentiation and optimization

- Chapter 18. Integrals

1 Volumes and the Riemann sums 2 Properties of the Riemann sums 3 The Riemann integral over rectangles 4 The variable density and the weight as the 3d Riemann integral 5 Ascending the dimensions: lengths, areas, volumes, and beyond 6 Outside the sandbox 7 The n-dimensional case 8 The center of mass 9 Change of variables

- Chapter 19. Vector fields

1 Overview of functions 2 Vector fields 3 The algebra and geometry of vector fields 4 The derivative of a function of several variables 5 The Chain Rule 6 What does the gradient tell us about the function? 7 Monotonicity 8 Differentiation 9 What vector fields are gradients?

- Chapter 20. Transformations

## 5 Part IV: Differential equations

- Chapter 21. Ordinary differential equations

1 Ordinary differential equations 2 Discrete models and setting up ODEs 3 Solution sets of ODEs 4 Change of variables in ODEs 5 Separation of variables in ODEs 6 The method of integrating factors 7 Solving ODEs numerically: Euler's method 8 Qualitative analysis of ODEs 9 The accuracy of Euler's method 10 Exactness* 11 Linearization of ODEs 12 Solving ODEs with series 13 Motion under forces: ODEs of second order

- Chapter 22. Vector and complex variables

1 The emergence of complex numbers 2 The algebra of complex numbers 3 Classification of quadratic polynomials 4 Calculus of complex variable 5 Power series 6 Solving ODEs with power series 7 Fourier series* 8 Eigenvalues and eigenvectors 9 Classification of linear functions 10 Classification of linear functions, continued

- Chapter 23. Systems of ODEs

1 The predator-prey model 2 Qualitative analysis of the predator-prey model 3 Solving Lotka–Volterra equations 4 Vector fields and systems of ODEs 5 Euler's method on the plane 6 Qualitative analysis of systems of ODEs 7 The vector notation and linear systems 8 Classification of linear systems 9 Classification of linear systems, continued

- Chapter 24. Applications of ODEs

1 Pursuit curves 2 Linearization 3 Discrete models and setting up vector ODEs 4 Motion under forces: ODEs of second order 5 Motion under forces: vector ODE of second order 6 Planetary motion 7 The two- and three-body problems 8 Boundary value problems

- Chapter 25. Partial differential equations

- Appendix: Discrete calculus
- Appendix: Projects
- Appendix: What shape of sword is best for cutting? (a model project, Part III)
- Appendix: Calculus exercises: advanced