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Calculus Volume 1

Key Equations

Calculus Volume 1Key Equations
Table of contents
  1. Preface
  2. 1 Functions and Graphs
    1. Introduction
    2. 1.1 Review of Functions
    3. 1.2 Basic Classes of Functions
    4. 1.3 Trigonometric Functions
    5. 1.4 Inverse Functions
    6. 1.5 Exponential and Logarithmic Functions
    7. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  3. 2 Limits
    1. Introduction
    2. 2.1 A Preview of Calculus
    3. 2.2 The Limit of a Function
    4. 2.3 The Limit Laws
    5. 2.4 Continuity
    6. 2.5 The Precise Definition of a Limit
    7. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  4. 3 Derivatives
    1. Introduction
    2. 3.1 Defining the Derivative
    3. 3.2 The Derivative as a Function
    4. 3.3 Differentiation Rules
    5. 3.4 Derivatives as Rates of Change
    6. 3.5 Derivatives of Trigonometric Functions
    7. 3.6 The Chain Rule
    8. 3.7 Derivatives of Inverse Functions
    9. 3.8 Implicit Differentiation
    10. 3.9 Derivatives of Exponential and Logarithmic Functions
    11. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  5. 4 Applications of Derivatives
    1. Introduction
    2. 4.1 Related Rates
    3. 4.2 Linear Approximations and Differentials
    4. 4.3 Maxima and Minima
    5. 4.4 The Mean Value Theorem
    6. 4.5 Derivatives and the Shape of a Graph
    7. 4.6 Limits at Infinity and Asymptotes
    8. 4.7 Applied Optimization Problems
    9. 4.8 L’Hôpital’s Rule
    10. 4.9 Newton’s Method
    11. 4.10 Antiderivatives
    12. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  6. 5 Integration
    1. Introduction
    2. 5.1 Approximating Areas
    3. 5.2 The Definite Integral
    4. 5.3 The Fundamental Theorem of Calculus
    5. 5.4 Integration Formulas and the Net Change Theorem
    6. 5.5 Substitution
    7. 5.6 Integrals Involving Exponential and Logarithmic Functions
    8. 5.7 Integrals Resulting in Inverse Trigonometric Functions
    9. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  7. 6 Applications of Integration
    1. Introduction
    2. 6.1 Areas between Curves
    3. 6.2 Determining Volumes by Slicing
    4. 6.3 Volumes of Revolution: Cylindrical Shells
    5. 6.4 Arc Length of a Curve and Surface Area
    6. 6.5 Physical Applications
    7. 6.6 Moments and Centers of Mass
    8. 6.7 Integrals, Exponential Functions, and Logarithms
    9. 6.8 Exponential Growth and Decay
    10. 6.9 Calculus of the Hyperbolic Functions
    11. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  8. A | Table of Integrals
  9. B | Table of Derivatives
  10. C | Review of Pre-Calculus
  11. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
  12. Index

Key Equations

Area between two curves, integrating on the x-axis A=∫ab[f(x)−g(x)]dxA=∫ab[f(x)−g(x)]dx
Area between two curves, integrating on the y-axis A=∫cd[u(y)−v(y)]dyA=∫cd[u(y)−v(y)]dy
Disk Method along the x-axis V=∫abπ[f(x)]2dxV=∫abπ[f(x)]2dx
Disk Method along the y-axis V=∫cdπ[g(y)]2dyV=∫cdπ[g(y)]2dy
Washer Method V=∫abπ[(f(x))2−(g(x))2]dxV=∫abπ[(f(x))2−(g(x))2]dx
Method of Cylindrical Shells V=∫ab(2πxf(x))dxV=∫ab(2πxf(x))dx
Arc Length of a Function of x Arc Length=∫ab1+[f′(x)]2dxArc Length=∫ab1+[f′(x)]2dx
Arc Length of a Function of y Arc Length=∫cd1+[g′(y)]2dyArc Length=∫cd1+[g′(y)]2dy
Surface Area of a Function of x Surface Area=∫ab(2πf(x)1+(f′(x))2)dxSurface Area=∫ab(2πf(x)1+(f′(x))2)dx
Mass of a one-dimensional object m=∫abρ(x)dxm=∫abρ(x)dx
Mass of a circular object m=∫0r2πxρ(x)dxm=∫0r2πxρ(x)dx
Work done on an object W=∫abF(x)dxW=∫abF(x)dx
Hydrostatic force on a plate F=∫abρw(x)s(x)dxF=∫abρw(x)s(x)dx
Mass of a lamina m=ρ∫abf(x)dxm=ρ∫abf(x)dx
Moments of a lamina Mx=ρ∫ab[f(x)]22dxandMy=ρ∫abxf(x)dxMx=ρ∫ab[f(x)]22dxandMy=ρ∫abxf(x)dx
Center of mass of a lamina x–=Mymandy–=Mxmx–=Mymandy–=Mxm
Natural logarithm function lnx=∫1x1tdtlnx=∫1x1tdt Z
Exponential function y=exy=ex lny=ln(ex)=xlny=ln(ex)=x Z
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