Calculus with Julia (https://jverzani.github.io/CalculusWithJuliaNotes.jl/basics.html) (site_libs/quarto-html/tippy.css) (site_libs/quarto-html/quarto-syntax-highlighting-7f8f88aac4f3542376d5c11b86a4c14d.css) (site_libs/quarto-html/quarto-syntax-highlighting-dark-d0ae9245876894da5ac7e18953ecc5cc.css) (site_libs/quarto-html/quarto-syntax-highlighting-7f8f88aac4f3542376d5c11b86a4c14d.css) (site_libs/bootstrap/bootstrap-icons.css) (site_libs/bootstrap/bootstrap-c0fec8dc2dddb07fb65bdfeb8e6a8e1d.min.css) (site_libs/bootstrap/bootstrap-dark-294ee5c3a98b849f5d3a98121b7162f1.min.css) (site_libs/bootstrap/bootstrap-c0fec8dc2dddb07fb65bdfeb8e6a8e1d.min.css) (site_libs/quarto-contrib/fontawesome-7-free-7.2.0/fontawesome.css) (styles.css)   (https://jverzani.github.io/CalculusWithJuliaNotes.jl/index.html) () ()  (https://jverzani.github.io/CalculusWithJuliaNotes.jl/index.html) Calculus with Julia   (https://github.com/jverzani/CalculusWithJuliaNotes.jl/tree/main/quarto/) (Source Code)   (index.html) (Toggle dark mode)    (Search)      (https://jverzani.github.io/CalculusWithJuliaNotes.jl/index.html) Preface        (https://jverzani.github.io/CalculusWithJuliaNotes.jl/index.html) Preface    (https://jverzani.github.io/CalculusWithJuliaNotes.jl/basics.html) Mathematical basics     (https://jverzani.github.io/CalculusWithJuliaNotes.jl/basics/calculator.html) 1 From calculator to computer     (https://jverzani.github.io/CalculusWithJuliaNotes.jl/basics/variables.html) 2 Variables     (https://jverzani.github.io/CalculusWithJuliaNotes.jl/basics/numbers_types.html) 3 Number systems     (https://jverzani.github.io/CalculusWithJuliaNotes.jl/basics/logical_expressions.html) 4 Inequalities, Logical expressions     (https://jverzani.github.io/CalculusWithJuliaNotes.jl/basics/vectors.html) 5 Points, vectors, and containers     (https://jverzani.github.io/CalculusWithJuliaNotes.jl/basics/ranges.html) 6 Ranges and sets       (https://jverzani.github.io/CalculusWithJuliaNotes.jl/precalc.html) Precalculus Concepts     (https://jverzani.github.io/CalculusWithJuliaNotes.jl/precalc/functions.html) 7 Functions     (https://jverzani.github.io/CalculusWithJuliaNotes.jl/precalc/plotting.html) 8 The graph of a function     (https://jverzani.github.io/CalculusWithJuliaNotes.jl/precalc/transformations.html) 9 Function transformations     (https://jverzani.github.io/CalculusWithJuliaNotes.jl/precalc/inversefunctions.html) 10 The inverse of a function     (https://jverzani.github.io/CalculusWithJuliaNotes.jl/precalc/polynomial.html) 11 Polynomials     (https://jverzani.github.io/CalculusWithJuliaNotes.jl/precalc/polynomial_roots.html) 12 Roots of a polynomial     (https://jverzani.github.io/CalculusWithJuliaNotes.jl/precalc/polynomials_package.html) 13 The Polynomials package     (https://jverzani.github.io/CalculusWithJuliaNotes.jl/precalc/rational_functions.html) 14 Rational functions     (https://jverzani.github.io/CalculusWithJuliaNotes.jl/precalc/exp_log_functions.html) 15 Exponential and logarithmic functions     (https://jverzani.github.io/CalculusWithJuliaNotes.jl/precalc/trig_functions.html) 16 Trigonometric functions     (https://jverzani.github.io/CalculusWithJuliaNotes.jl/precalc/julia_overview.html) 17 Overview of Julia commands       (https://jverzani.github.io/CalculusWithJuliaNotes.jl/limits.html) Limits and continuity     (https://jverzani.github.io/CalculusWithJuliaNotes.jl/limits/limits.html) 18 Limits     (https://jverzani.github.io/CalculusWithJuliaNotes.jl/limits/limits_extensions.html) 19 Limits, issues, extensions of the concept     (https://jverzani.github.io/CalculusWithJuliaNotes.jl/limits/continuity.html) 20 Continuity     (https://jverzani.github.io/CalculusWithJuliaNotes.jl/limits/intermediate_value_theorem.html) 21 Implications of continuity     (https://jverzani.github.io/CalculusWithJuliaNotes.jl/limits/sequences_series.html) 22 Sequences and series       (https://jverzani.github.io/CalculusWithJuliaNotes.jl/derivatives.html) Derivatives     (https://jverzani.github.io/CalculusWithJuliaNotes.jl/derivatives/derivatives.html) 23 Derivatives     (https://jverzani.github.io/CalculusWithJuliaNotes.jl/derivatives/numeric_derivatives.html) 24 Computing derivatives in Julia     (https://jverzani.github.io/CalculusWithJuliaNotes.jl/derivatives/symbolic_derivatives.html) 25 Symbolic derivatives     (https://jverzani.github.io/CalculusWithJuliaNotes.jl/derivatives/mean_value_theorem.html) 26 Implications of Differentiability     (https://jverzani.github.io/CalculusWithJuliaNotes.jl/derivatives/optimization.html) 27 Optimization     (https://jverzani.github.io/CalculusWithJuliaNotes.jl/derivatives/first_second_derivatives.html) 28 The first and second derivatives     (https://jverzani.github.io/CalculusWithJuliaNotes.jl/derivatives/curve_sketching.html) 29 Curve sketching     (https://jverzani.github.io/CalculusWithJuliaNotes.jl/derivatives/linearization.html) 30 Linearization     (https://jverzani.github.io/CalculusWithJuliaNotes.jl/derivatives/newtons_method.html) 31 Newton’s method     (https://jverzani.github.io/CalculusWithJuliaNotes.jl/derivatives/more_zeros.html) 32 Other zero-finding algorithms     (https://jverzani.github.io/CalculusWithJuliaNotes.jl/derivatives/lhospitals_rule.html) 33 L’Hospital’s rule     (https://jverzani.github.io/CalculusWithJuliaNotes.jl/derivatives/implicit_differentiation.html) 34 Implicit differentiation     (https://jverzani.github.io/CalculusWithJuliaNotes.jl/derivatives/related_rates.html) 35 Related rates     (https://jverzani.github.io/CalculusWithJuliaNotes.jl/derivatives/taylor_series_polynomials.html) 36 Taylor polynomials, series, and approximating polynomials       (https://jverzani.github.io/CalculusWithJuliaNotes.jl/integrals.html) Integrals     (https://jverzani.github.io/CalculusWithJuliaNotes.jl/integrals/area.html) 37 Area under a curve     (https://jverzani.github.io/CalculusWithJuliaNotes.jl/integrals/ftc.html) 38 Fundamental Theorem of Calculus     (https://jverzani.github.io/CalculusWithJuliaNotes.jl/integrals/numeric_integrals.html) 39 Numeric approximations to definite integrals     (https://jverzani.github.io/CalculusWithJuliaNotes.jl/integrals/substitution.html) 40 Substitution     (https://jverzani.github.io/CalculusWithJuliaNotes.jl/integrals/integration_by_parts.html) 41 Integration by parts     (https://jverzani.github.io/CalculusWithJuliaNotes.jl/integrals/partial_fractions.html) 42 Partial fractions     (https://jverzani.github.io/CalculusWithJuliaNotes.jl/integrals/area_between_curves.html) 43 Area between curves     (https://jverzani.github.io/CalculusWithJuliaNotes.jl/integrals/improper_integrals.html) 44 Improper integrals     (https://jverzani.github.io/CalculusWithJuliaNotes.jl/integrals/mean_value_theorem.html) 45 Mean value theorem for integrals     (https://jverzani.github.io/CalculusWithJuliaNotes.jl/integrals/center_of_mass.html) 46 Center of mass     (https://jverzani.github.io/CalculusWithJuliaNotes.jl/integrals/arc_length.html) 47 Arc length     (https://jverzani.github.io/CalculusWithJuliaNotes.jl/integrals/volumes_slice.html) 48 Volumes by slicing     (https://jverzani.github.io/CalculusWithJuliaNotes.jl/integrals/surface_area.html) 49 Surface area     (https://jverzani.github.io/CalculusWithJuliaNotes.jl/integrals/orthogonal_polynomials.html) 50 Orthogonal polynomials     (https://jverzani.github.io/CalculusWithJuliaNotes.jl/integrals/twelve-qs.html) 51 A dozen minima for a parabola       (https://jverzani.github.io/CalculusWithJuliaNotes.jl/ODEs.html) ODEs     (https://jverzani.github.io/CalculusWithJuliaNotes.jl/ODEs/odes.html) 52 ODEs     (https://jverzani.github.io/CalculusWithJuliaNotes.jl/ODEs/euler.html) 53 Euler’s method     (https://jverzani.github.io/CalculusWithJuliaNotes.jl/ODEs/solve.html) 54 The problem-algorithm-solve interface     (https://jverzani.github.io/CalculusWithJuliaNotes.jl/ODEs/differential_equations.html) 55 The DifferentialEquations suite       (https://jverzani.github.io/CalculusWithJuliaNotes.jl/differentiable_vector_calculus.html) Differential vector calculus     (https://jverzani.github.io/CalculusWithJuliaNotes.jl/differentiable_vector_calculus/polar_coordinates.html) 56 Polar coordinates and curves     (https://jverzani.github.io/CalculusWithJuliaNotes.jl/differentiable_vector_calculus/vectors.html) 57 Vectors and matrices     (https://jverzani.github.io/CalculusWithJuliaNotes.jl/differentiable_vector_calculus/vector_valued_functions.html) 58 Vector-valued functions, \(f:R \rightarrow R^n\)      (https://jverzani.github.io/CalculusWithJuliaNotes.jl/differentiable_vector_calculus/scalar_functions.html) 59 Scalar functions     (https://jverzani.github.io/CalculusWithJuliaNotes.jl/differentiable_vector_calculus/scalar_functions_applications.html) 60 Applications with scalar functions     (https://jverzani.github.io/CalculusWithJuliaNotes.jl/differentiable_vector_calculus/vector_fields.html) 61 Functions \(R^n \rightarrow R^m\)      (https://jverzani.github.io/CalculusWithJuliaNotes.jl/differentiable_vector_calculus/matrix_calculus_notes.html) 62 Matrix calculus     (https://jverzani.github.io/CalculusWithJuliaNotes.jl/differentiable_vector_calculus/plots_plotting.html) 63 2D and 3D plots in Julia with Plots       (https://jverzani.github.io/CalculusWithJuliaNotes.jl/integral_vector_calculus.html) Integral vector calculus     (https://jverzani.github.io/CalculusWithJuliaNotes.jl/integral_vector_calculus/double_triple_integrals.html) 64 Multi-dimensional integrals     (https://jverzani.github.io/CalculusWithJuliaNotes.jl/integral_vector_calculus/line_integrals.html) 65 Line and surface integrals     (https://jverzani.github.io/CalculusWithJuliaNotes.jl/integral_vector_calculus/div_grad_curl.html) 66 The gradient, divergence, and curl     (https://jverzani.github.io/CalculusWithJuliaNotes.jl/integral_vector_calculus/stokes_theorem.html) 67 Green’s theorem, Stokes’ theorem, and the divergence theorem     (https://jverzani.github.io/CalculusWithJuliaNotes.jl/integral_vector_calculus/review.html) 68 Quick review of vector calculus       (https://jverzani.github.io/CalculusWithJuliaNotes.jl/alternatives.html) Alternative packages     (https://jverzani.github.io/CalculusWithJuliaNotes.jl/alternatives/symbolics.html) 69 Symbolics.jl     (https://jverzani.github.io/CalculusWithJuliaNotes.jl/alternatives/SciML.html) 70 The SciML suite of packages     (https://jverzani.github.io/CalculusWithJuliaNotes.jl/alternatives/plotly_plotting.html) 71 JavaScript based plotting libraries     (https://jverzani.github.io/CalculusWithJuliaNotes.jl/alternatives/makie_plotting.html) 72 Calculus plots with Makie       Appendices     (https://jverzani.github.io/CalculusWithJuliaNotes.jl/misc/getting_started_with_julia.html) 73 Getting started with Julia     (https://jverzani.github.io/CalculusWithJuliaNotes.jl/misc/julia_interfaces.html) 74 Julia interfaces     (https://jverzani.github.io/CalculusWithJuliaNotes.jl/misc/calculus_with_julia.html) 75 The CalculusWithJulia package     (https://jverzani.github.io/CalculusWithJuliaNotes.jl/misc/unicode.html) 76 Usages of Unicode symbols     (https://jverzani.github.io/CalculusWithJuliaNotes.jl/misc/quick_notes.html) 77 Quick introduction to Calculus with Julia       (https://jverzani.github.io/CalculusWithJuliaNotes.jl/references.html) References        Table of contents (index.html#preface) Preface (index.html#running-julia) Running Julia     (https://github.com/jverzani/CalculusWithJuliaNotes.jl/edit/main/quarto/index.qmd)  Edit this page  (https://github.com/jverzani/CalculusWithJuliaNotes.jl/issues/new)  Report an issue      Calculus with Julia  Author John Verzani    Published August 31, 2026     Preface  Calculus with Julia   This is a set of notes for learning (http://en.wikipedia.org/wiki/Calculus) calculus using the (https://julialang.org) Julia  language. Julia is an open-source programming language with an easy to learn syntax that is well suited for this task. Read “(https://jverzani.github.io/CalculusWithJuliaNotes.jl/misc/getting_started_with_julia.html) Getting started with Julia ” to learn how to install and customize Julia for following along with these notes. Read “(https://jverzani.github.io/CalculusWithJuliaNotes.jl/misc/julia_interfaces.html) Julia interfaces ” to review different ways to interact with a Julia installation. Since the mid 90s there has been a push to teach calculus using many different points of view. The (http://www.math.harvard.edu/~knill/pedagogy/harvardcalculus/) Harvard style rule of four says that as much as possible the conversation should include a graphical, numerical, algebraic, and verbal component. These notes use the programming language (http://julialang.org) Julia to illustrate the graphical, numerical, and, at times, the algebraic aspects of calculus. There are many examples of integrating a computer algebra system (such as Mathematica , Maple , or Sage ) into the calculus conversation. Computer algebra systems can be magical. The popular (http://www.wolframalpha.com/) WolframAlpha website calls the full power of Mathematica while allowing an informal syntax that is flexible enough to be used as a backend for Apple’s Siri feature. (“Siri what is the graph of x squared minus 4?”) For learning purposes, computer algebra systems model very well the algebraic/symbolic treatment of the material while providing means to illustrate the numeric aspects. These notes are a bit different in that Julia is primarily used for the numeric style of computing and the algebraic/symbolic treatment is added on. Doing the symbolic treatment by hand can be very beneficial while learning, and computer algebra systems make those exercises seem kind of redundant, as the finished product can be produced much more easily. Our real goal is to get at the concepts using technology as much as possible without getting bogged down in the mechanics of the computer language. We feel Julia has a very natural syntax that makes the initial start up not so much more difficult than using a calculator, but with a language that has a tremendous upside. The notes restrict themselves to a reduced set of computational concepts. This set is sufficient for working many of the problems in calculus, but do not cover thoroughly many aspects of programming. (Those who are interested can go off on their own and Julia provides a rich opportunity to do so.) Within this restricted set, are operators that make many of the computations of calculus reduce to a function call of the form action(function, arguments...) . With a small collection of actions that can be composed, many of the problems associated with introductory calculus can be attacked. These notes are presented in pages covering a fairly focused concept, in a spirit similar to a section of a book. Just like a book, there are try-it-yourself questions at the end of each page. All have a limited number of self-graded answers. These notes borrow ideas from many sources, for example Strang ((https://jverzani.github.io/CalculusWithJuliaNotes.jl/references.html#ref-Strang) n.d. ) , Knill ((https://jverzani.github.io/CalculusWithJuliaNotes.jl/references.html#ref-Knill) n.d. ) , Schey ((https://jverzani.github.io/CalculusWithJuliaNotes.jl/references.html#ref-Schey) 1997 ) , Hass et al. ((https://jverzani.github.io/CalculusWithJuliaNotes.jl/references.html#ref-Thomas) 2018 ) , Rogawski et al. ((https://jverzani.github.io/CalculusWithJuliaNotes.jl/references.html#ref-RogawskiAdams) 2019 ) , Angenent ((https://jverzani.github.io/CalculusWithJuliaNotes.jl/references.html#ref-Angenent) 2012 ) , several Wikipedia pages, and other sources. These notes are accompanied by a Julia package CalculusWithJulia that provides some simple functions to streamline some common tasks and loads some useful packages that will be used repeatedly. These notes are presented as a Quarto book. To learn more about Quarto books visit (https://quarto.org/docs/books) https://quarto.org/docs/books . Compiling these notes into a pdf file with Quarto is possible with some piecing together of parts. The result is less than optimal and also quite large. (https://www.dropbox.com/scl/fi/vv8ee9a5xb9qbsw87f3bl/CalculusWithJulia.pdf?rlkey=u7ambk19sowelxtgiaosjozzo&st=i0hxwx29&dl=0) Download pdf version  To contribute – say by suggesting additional topics, correcting a mistake, or fixing a typo – click the “Edit this page” link and join the list of (https://github.com/jverzani/CalculusWithJuliaNotes.jl/graphs/contributors) contributors . Thanks to all contributors. A very special thanks goes out to @fangliu-tju for their careful and most-appreciated proofreading and error spotting spread over a series of PRs. Running Julia For individual users Julia is installed quite easily with the juliaup utility or by other means. There are some brief installation notes in the overview of Julia commands. Calculus with Julia version 0.27, produced on August 31, 2026. Angenent, Sigurd. 2012. Wisconsin Calculus . Https://github.com/SigurdAngenent/WisconsinCalculus/tree/master.  Hass, Joel R., Christopher E. Heil, and Maurice D. Weir. 2018. Thomas’ Calculus . Pearson.  Knill, Oliver. n.d. Some Teaching Notes . (https://people.math.harvard.edu/~knill/teach/index.html) https://people.math.harvard.edu/~knill/teach/index.html .  Rogawski, Jon, Colin Adams, and Robert Franzosa. 2019. Calculus . Macmillan.  Schey, H. M. 1997. Div, Grad, Curl, and All That . W.W. Norton.  Strang, Gilbert. n.d. Calculus . Wellesley-Cambridge Press. (https://ocw.mit.edu/courses/res-18-001-calculus-online-textbook-spring-2005/) https://ocw.mit.edu/courses/res-18-001-calculus-online-textbook-spring-2005/ .       (https://jverzani.github.io/CalculusWithJuliaNotes.jl/basics.html) Mathematical basics       Copyright 2022-26, John Verzani (https://github.com/jverzani/CalculusWithJuliaNotes.jl/edit/main/quarto/index.qmd)  Edit this page  (https://github.com/jverzani/CalculusWithJuliaNotes.jl/issues/new)  Report an issue