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

[bcrowell]

• Author: Sigurd Angenent, Laurentiu Maxim, and Joel Robbin
• Title: UW Calculus
• Download Link: http://www.math.wisc.edu/~angenent/Free-Lecture-Notes/
• Prerequisities:
• Contents:

Table of Contents:

[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 [itex]a^x[/itex] and the definition of [itex]e[/itex]
[*] Derivatives of Logarithms 
[*] Limits involving exponentials and logarithms
[*] Exponential growth and decay
[*] Exercises
[/LIST]
[*] The Integral
[LIST]
[*] Area under a Graph
[*] When [itex]f[/itex] 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]
[*] Answers and Hints
[/LIST]

[LIST]
[*] Methods of Integration
[LIST]
[*] The indefinite integral
[*] You can always check the answer
[*] About “+C”
[*] 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 [itex]x\rightarrow 0[/itex], keeping [itex]n[/itex] fixed
[*] The limit [itex]n\rightarrow \infty[/itex], keeping [itex]x[/itex] fixed
[*] Convergence of Taylor Series
[*] Leibniz’ formulas for [itex]\ln 2[/itex] and [itex]\pi/4[/itex]
[*] 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 [itex]\vec{x}^\prime (t)[/itex] 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]

 

[bcrowell]

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.