# UW Calculus by Sigurd Angenent, Laurentiu Maxim, and Joel Robbin

• Calculus
• bcrowell
In summary, "UW Calculus" by Sigurd Angenent, Laurentiu Maxim, and Joel Robbin is a comprehensive text for a two-semester freshman calculus course. It covers topics such as numbers and functions, derivatives, limits and continuous functions, graph sketching and max-min problems, exponentials and logarithms, integrals, applications of the integral, methods of integration, Taylor's formula and infinite series, complex numbers and the complex exponential, differential equations, vectors, and vector functions and parametrized curves. The book is freely available online and includes a variety of problem sets and applications. It is licensed under the open-source GFDL license.

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Code:
[LIST]
[*] Numbers and Functions
[LIST]
[*] What is a number?
[*] Exercises
[*] Functions
[*] Inverse functions and Implicit functions
[*] Exercises
[/LIST]
[*] Derivatives (1)
[LIST]
[*] The tangent to a curve
[*] An example - tangent to a parabola
[*] Instantaneous velocity
[*] Rates of change
[*] Examples of rates of change
[*] Exercises
[/LIST]
[*] Limits and Continuous Function
[LIST]
[*] Informal denition of limits
[*] The formal, authoritative, definition of limit
[*] Exercises
[*] Variations on the limit theme
[*] Properties of the Limit
[*] Examples of limit computations
[*] When limits fail to exist
[*] What's in a name?
[*] Limits and Inequalities
[*] Continuity
[*] Substitution in Limits
[*] Exercises
[*] Two Limits in Trigonometry
[*] Exercises
[/LIST]
[*] Derivatives (2)
[LIST]
[*] Derivatives Defined
[*] Direct computation of derivatives
[*] Differentiable implies Continuous
[*] Some non-differentiable functions
[*] Exercises
[*] The Differentiation Rules
[*] Differentiating powers of functions
[*] Exercises
[*] Higher Derivatives
[*] Exercises
[*] Differentiating Trigonometric functions
[*] Exercises
[*] The Chain Rule
[*] Exercises
[*] Implicit dierentiation
[*] Exercises
[/LIST]
[*] Graph Sketching and Max-Min Problems
[LIST]
[*] Tangent and Normal lines to a graph
[*] The Intermediate Value Theorem
[*] Exercises
[*] Finding sign changes of a function
[*] Increasing and decreasing functions
[*] Examples
[*] Maxima and Minima
[*] Must there always be a maximum?
[*] Examples - functions with and without maxima or minima
[*] General method for sketching the graph of a function
[*] Convexity, Concavity and the Second Derivative
[*] Proofs of some of the theorems
[*] Exercises
[*] Optimization Problems
[*] Exercises
[/LIST]
[*] Exponentials and Logarithms (naturally)
[LIST]
[*] Exponents
[*] Logarithms
[*] Properties of logarithms
[*] Graphs of exponential functions and logarithms
[*] The derivative of $a^x$ and the definition of $e$
[*] Derivatives of Logarithms
[*] Limits involving exponentials and logarithms
[*] Exponential growth and decay
[*] Exercises
[/LIST]
[*] The Integral
[LIST]
[*] Area under a Graph
[*] When $f$ changes its sign
[*] The Fundamental Theorem of Calculus
[*] Exercises
[*] The indefinite integral
[*] Properties of the Integral
[*] The definite integral as a function of its integration bounds
[*] Method of substitution
[*] Exercises
[/LIST]
[*] Applications of the integral
[LIST]
[*] Areas between graphs
[*] Exercises
[*] Cavalieri's principle and volumes of solids
[*] Examples of volumes of solids of revolution
[*] Volumes by cylindrical shells
[*] Exercises
[*] Distance from velocity, velocity from acceleration
[*] The length of a curve
[*] Examples of length computations
[*] Exercises
[*] Work done by a force
[*] Work done by an electric current
[/LIST]
[/LIST]

Code:
[LIST]
[*] Methods of Integration
[LIST]
[*] The indefinite integral
[*] You can always check the answer
[*] Standard Integrals
[*] Method of substitution
[*] The double angle trick
[*] Integration by Parts
[*] Reduction Formulas
[*] Partial Fraction Expansion
[*] Problems
[/LIST]
[*] Taylor’s Formula and Infinite Series
[LIST]
[*] Taylor Polynomials
[*] Examples
[*] Some special Taylor polynomials
[*] The Remainder Term
[*] Lagrange’s Formula for the Remainder Term
[*] The limit as $x\rightarrow 0$, keeping $n$ fixed
[*] The limit $n\rightarrow \infty$, keeping $x$ fixed
[*] Convergence of Taylor Series
[*] Leibniz’ formulas for $\ln 2$ and $\pi/4$
[*] Proof of Lagrange’s formula
[*] Proof of Theorem 16.8
[*] Problems
[/LIST]
[*] Complex Numbers and the Complex Exponential
[LIST]
[*] Complex numbers
[*] Argument and Absolute Value
[*] Geometry of Arithmetic
[*] Applications in Trigonometry
[*] Calculus of complex valued functions
[*] The Complex Exponential Function
[*] Complex solutions of polynomial equations
[*] Other handy things you can do with complex numbers
[*] Problems
[/LIST]
[*] Differential Equations
[LIST]
[*] What is a DiffEq?
[*] First Order Separable Equations
[*] First Order Linear Equations
[*] Dynamical Systems and Determinism
[*] Higher order equations
[*] Constant Coefficient Linear Homogeneous Equations
[*] Inhomogeneous Linear Equations
[*] Variation of Constants
[*] Applications of Second Order Linear Equations
[*] Problems
[/LIST]
[*] Vectors
[LIST]
[*] Introduction to vectors
[*] Parametric equations for lines and planes
[*] Vector Bases
[*] Dot Product
[*] Cross Product
[*] A few applications of the cross product
[*] Notation
[*] Problems
[/LIST]
[*] Vector Functions and Parametrized Curves
[LIST]
[*] Parametric Curves
[*] Examples of parametrized curves
[*] The derivative of a vector function
[*] Higher derivatives and product rules
[*] Interpretation of $\vec{x}^\prime (t)$ as the velocity vector
[*] Acceleration and Force
[*] Tangents and the unit tangent vector
[*] Sketching a parametric curve
[*] Length of a curve
[*] The arclength function
[*] Graphs in Cartesian and in Polar Coordinates
[*] Problems
[/LIST]
[/LIST]

Last edited by a moderator:
This is a text for a two-semester freshman calculus course, with a typical approach and order of topics. There are good problems sets, including word problems and applications. There's a nice looking layout with many figures. The book is free online, and is under the open-source GFDL license.

## 1. What topics does "UW Calculus" cover?

The book covers topics in single variable calculus, multivariable calculus, and vector calculus. It also includes applications to physics, engineering, and other fields.

## 2. Who are the authors of "UW Calculus"?

The authors of "UW Calculus" are Sigurd Angenent, Laurentiu Maxim, and Joel Robbin, all of whom are professors in the Mathematics Department at the University of Wisconsin-Madison.

## 3. Is "UW Calculus" suitable for beginners?

While "UW Calculus" assumes some prior knowledge of basic algebra and functions, it is designed to be accessible to beginners. The authors provide clear explanations and examples throughout the book to help readers understand the concepts.

## 4. Are there any resources available to supplement the book?

Yes, there are online resources such as lecture videos and practice problems available on the authors' website. There is also a discussion forum where readers can ask questions and interact with the authors and other readers.

## 5. Is "UW Calculus" used in any university courses?

Yes, "UW Calculus" is used as the textbook for several calculus courses at the University of Wisconsin-Madison, including Math 221, 222, and 234. It is also used in other universities around the world.

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