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

Calculus Illustrated

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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.

1 Part R: Review of 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

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 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

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 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

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

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

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

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

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

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?

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 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

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 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