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Calculus Volume 2

B | Table of Derivatives

Calculus Volume 2B | Table of Derivatives
  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. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. 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. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. 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. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. 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. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. 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. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. 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. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. 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. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. 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

General Formulas

1. ddx(c)=0ddx(c)=0

2. ddx(f(x)+g(x))=f(x)+g(x)ddx(f(x)+g(x))=f(x)+g(x)

3. ddx(f(x)g(x))=f(x)g(x)+f(x)g(x)ddx(f(x)g(x))=f(x)g(x)+f(x)g(x)

4. ddx(xn)=nxn1,for real numbersnddx(xn)=nxn1,for real numbersn

5. ddx(cf(x))=cf(x)ddx(cf(x))=cf(x)

6. ddx(f(x)g(x))=f(x)g(x)ddx(f(x)g(x))=f(x)g(x)

7. ddx(f(x)g(x))=g(x)f(x)f(x)g(x)(g(x))2ddx(f(x)g(x))=g(x)f(x)f(x)g(x)(g(x))2

8. ddx[f(g(x))]=f(g(x))·g(x)ddx[f(g(x))]=f(g(x))·g(x)

Trigonometric Functions

9. ddx(sinx)=cosxddx(sinx)=cosx

10. ddx(tanx)=sec2xddx(tanx)=sec2x

11. ddx(secx)=secxtanxddx(secx)=secxtanx

12. ddx(cosx)=sinxddx(cosx)=sinx

13. ddx(cotx)=csc2xddx(cotx)=csc2x

14. ddx(cscx)=−cscxcotxddx(cscx)=−cscxcotx

Inverse Trigonometric Functions

15. ddx(sin−1x)=11x2ddx(sin−1x)=11x2

16. ddx(tan−1x)=11+x2ddx(tan−1x)=11+x2

17. ddx(sec−1x)=1|x|x21ddx(sec−1x)=1|x|x21

18. ddx(cos−1x)=11x2ddx(cos−1x)=11x2

19. ddx(cot−1x)=11+x2ddx(cot−1x)=11+x2

20. ddx(csc−1x)=1|x|x21ddx(csc−1x)=1|x|x21

Exponential and Logarithmic Functions

21. ddx(ex)=exddx(ex)=ex

22. ddx(ln|x|)=1xddx(ln|x|)=1x

23. ddx(bx)=bxlnbddx(bx)=bxlnb

24. ddx(logbx)=1xlnbddx(logbx)=1xlnb

Hyperbolic Functions

25. ddx(sinhx)=coshxddx(sinhx)=coshx

26. ddx(tanhx)=sech2xddx(tanhx)=sech2x

27. ddx(sechx)=−sechxtanhxddx(sechx)=−sechxtanhx

28. ddx(coshx)=sinhxddx(coshx)=sinhx

29. ddx(cothx)=csch2xddx(cothx)=csch2x

30. ddx(cschx)=−cschxcothxddx(cschx)=−cschxcothx

Inverse Hyperbolic Functions

31. ddx(sinh−1x)=1x2+1ddx(sinh−1x)=1x2+1

32. ddx(tanh−1x)=11x2(|x|<1)ddx(tanh−1x)=11x2(|x|<1)

33. ddx(sech−1x)=1x1x2(0<x<1)ddx(sech−1x)=1x1x2(0<x<1)

34. ddx(cosh−1x)=1x21(x>1)ddx(cosh−1x)=1x21(x>1)

35. ddx(coth−1x)=11x2(|x|>1)ddx(coth−1x)=11x2(|x|>1)

36. ddx(csch−1x)=1|x|1+x2(x0)ddx(csch−1x)=1|x|1+x2(x0)

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