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

A draft in progress...

## Contents

## 1 Part R: Review of functions

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

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

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

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

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

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...

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

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

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 Exponential models 7 Motion on the plane 8 Functions of several variables

## 3 Part II: Integral calculus

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*

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

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

1 From linear to quadratic approximations 2 Taylor polynomials 3 Sequences of functions 4 Infinite series 5 Examples of series 6 Properties of series 7 Calculus of power series 8 Fourier series

1 A ball is thrown... 2 Parametric curves on the plane 3 Polar coordinates 4 Differential equations 5 Functions of two variables 6 The centroid of a flat object 7 Vector fields 8 Differential forms 9 Discrete forms

## 4 Part III: Calculus in higher dimensions

- Functions in higher dimensions
- Differential calculus of parametric curves
- Functions of several variables: derivatives and integrals
- Line and surface integrals
- Fundamental theorems of calculus
- Examples and exercises: calculus in higher dimensions

## 5 Part IV: 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

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

1 Pursuit curves 2 Linearization 3 Motion under forces: ODEs of second order 4 Motion under forces: systems of second order 5 Planetary motion