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

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  1. Jan 20, 2013 #1

    bcrowell

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    Table of Contents:
    Code (Text):

    [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]
     
    Code (Text):

    [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]
     
     
    Last edited by a moderator: Jan 23, 2013
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  3. Jan 20, 2013 #2

    bcrowell

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    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.
     
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