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Calculus Volume 2

Key Equations

Calculus Volume 2Key Equations
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  1. Preface
  2. 1 Integration
    1. Introduction
    2. 1.1 Approximating Areas
    3. 1.2 The Definite Integral
    4. 1.3 The Fundamental Theorem of Calculus
    5. 1.4 Integration Formulas and the Net Change Theorem
    6. 1.5 Substitution
    7. 1.6 Integrals Involving Exponential and Logarithmic Functions
    8. 1.7 Integrals Resulting in Inverse Trigonometric Functions
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Chapter Review Exercises
  3. 2 Applications of Integration
    1. Introduction
    2. 2.1 Areas between Curves
    3. 2.2 Determining Volumes by Slicing
    4. 2.3 Volumes of Revolution: Cylindrical Shells
    5. 2.4 Arc Length of a Curve and Surface Area
    6. 2.5 Physical Applications
    7. 2.6 Moments and Centers of Mass
    8. 2.7 Integrals, Exponential Functions, and Logarithms
    9. 2.8 Exponential Growth and Decay
    10. 2.9 Calculus of the Hyperbolic Functions
    11. Key Terms
    12. Key Equations
    13. Key Concepts
    14. Chapter Review Exercises
  4. 3 Techniques of Integration
    1. Introduction
    2. 3.1 Integration by Parts
    3. 3.2 Trigonometric Integrals
    4. 3.3 Trigonometric Substitution
    5. 3.4 Partial Fractions
    6. 3.5 Other Strategies for Integration
    7. 3.6 Numerical Integration
    8. 3.7 Improper Integrals
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Chapter Review Exercises
  5. 4 Introduction to Differential Equations
    1. Introduction
    2. 4.1 Basics of Differential Equations
    3. 4.2 Direction Fields and Numerical Methods
    4. 4.3 Separable Equations
    5. 4.4 The Logistic Equation
    6. 4.5 First-order Linear Equations
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Chapter Review Exercises
  6. 5 Sequences and Series
    1. Introduction
    2. 5.1 Sequences
    3. 5.2 Infinite Series
    4. 5.3 The Divergence and Integral Tests
    5. 5.4 Comparison Tests
    6. 5.5 Alternating Series
    7. 5.6 Ratio and Root Tests
    8. Key Terms
    9. Key Equations
    10. Key Concepts
    11. Chapter Review Exercises
  7. 6 Power Series
    1. Introduction
    2. 6.1 Power Series and Functions
    3. 6.2 Properties of Power Series
    4. 6.3 Taylor and Maclaurin Series
    5. 6.4 Working with Taylor Series
    6. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Chapter Review Exercises
  8. 7 Parametric Equations and Polar Coordinates
    1. Introduction
    2. 7.1 Parametric Equations
    3. 7.2 Calculus of Parametric Curves
    4. 7.3 Polar Coordinates
    5. 7.4 Area and Arc Length in Polar Coordinates
    6. 7.5 Conic Sections
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Chapter Review Exercises
  9. A | Table of Integrals
  10. B | Table of Derivatives
  11. C | Review of Pre-Calculus
  12. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
  13. Index
  • Properties of Sigma Notation
    i=1nc=nci=1nc=nc
    i=1ncai=ci=1naii=1ncai=ci=1nai
    i=1n(ai+bi)=i=1nai+i=1nbii=1n(ai+bi)=i=1nai+i=1nbi
    i=1n(aibi)=i=1naii=1nbii=1n(aibi)=i=1naii=1nbi
    i=1nai=i=1mai+i=m+1naii=1nai=i=1mai+i=m+1nai
  • Sums and Powers of Integers
    i=1ni=1+2++n=n(n+1)2i=1ni=1+2++n=n(n+1)2
    i=1ni2=12+22++n2=n(n+1)(2n+1)6i=1ni2=12+22++n2=n(n+1)(2n+1)6
    i=0ni3=13+23++n3=n2(n+1)24i=0ni3=13+23++n3=n2(n+1)24
  • Left-Endpoint Approximation
    ALn=f(x0)Δx+f(x1)Δx++f(xn1)Δx=i=1nf(xi1)ΔxALn=f(x0)Δx+f(x1)Δx++f(xn1)Δx=i=1nf(xi1)Δx
  • Right-Endpoint Approximation
    ARn=f(x1)Δx+f(x2)Δx++f(xn)Δx=i=1nf(xi)ΔxARn=f(x1)Δx+f(x2)Δx++f(xn)Δx=i=1nf(xi)Δx
  • Definite Integral
    abf(x)dx=limni=1nf(xi*)Δxabf(x)dx=limni=1nf(xi*)Δx
  • Properties of the Definite Integral
    aaf(x)dx=0aaf(x)dx=0
    baf(x)dx=abf(x)dxbaf(x)dx=abf(x)dx
    ab[f(x)+g(x)]dx=abf(x)dx+abg(x)dxab[f(x)+g(x)]dx=abf(x)dx+abg(x)dx
    ab[f(x)g(x)]dx=abf(x)dxabg(x)dxab[f(x)g(x)]dx=abf(x)dxabg(x)dx
    abcf(x)dx=cabf(x)abcf(x)dx=cabf(x) for constant c
    abf(x)dx=acf(x)dx+cbf(x)dxabf(x)dx=acf(x)dx+cbf(x)dx
  • Mean Value Theorem for Integrals
    If f(x)f(x) is continuous over an interval [a,b],[a,b], then there is at least one point c[a,b]c[a,b] such that f(c)=1baabf(x)dx.f(c)=1baabf(x)dx.
  • Fundamental Theorem of Calculus Part 1
    If f(x)f(x) is continuous over an interval [a,b],[a,b], and the function F(x)F(x) is defined by F(x)=axf(t)dt,F(x)=axf(t)dt, then F(x)=f(x).F(x)=f(x).
  • Fundamental Theorem of Calculus Part 2
    If f is continuous over the interval [a,b][a,b] and F(x)F(x) is any antiderivative of f(x),f(x), then abf(x)dx=F(b)F(a).abf(x)dx=F(b)F(a).
  • Net Change Theorem
    F(b)=F(a)+abF'(x)dxF(b)=F(a)+abF'(x)dx or abF'(x)dx=F(b)F(a)abF'(x)dx=F(b)F(a)
  • Substitution with Indefinite Integrals
    f[g(x)]g(x)dx=f(u)du=F(u)+C=F(g(x))+Cf[g(x)]g(x)dx=f(u)du=F(u)+C=F(g(x))+C
  • Substitution with Definite Integrals
    abf(g(x))g'(x)dx=g(a)g(b)f(u)duabf(g(x))g'(x)dx=g(a)g(b)f(u)du
  • Integrals of Exponential Functions
    exdx=ex+Cexdx=ex+C
    axdx=axlna+Caxdx=axlna+C
  • Integration Formulas Involving Logarithmic Functions
    x−1dx=ln|x|+Cx−1dx=ln|x|+C
    lnxdx=xlnxx+C=x(lnx1)+Clnxdx=xlnxx+C=x(lnx1)+C
    logaxdx=xlna(lnx1)+Clogaxdx=xlna(lnx1)+C
  • Integrals That Produce Inverse Trigonometric Functions
    dua2u2=sin−1(ua)+Cdua2u2=sin−1(ua)+C
    dua2+u2=1atan−1(ua)+Cdua2+u2=1atan−1(ua)+C
    duuu2a2=1asec−1(ua)+Cduuu2a2=1asec−1(ua)+C
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