Skip to ContentGo to accessibility pageKeyboard shortcuts menu
OpenStax Logo
Calculus Volume 1

Key Equations

Calculus Volume 1Key Equations
Menu
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

Properties of Sigma Notation ∑i=1nc=nc∑i=1nc=nc
 ∑i=1ncai=c∑i=1nai∑i=1ncai=c∑i=1nai
 ∑i=1n(ai+bi)=∑i=1nai+∑i=1nbi∑i=1n(ai+bi)=∑i=1nai+∑i=1nbi
 ∑i=1n(ai−bi)=∑i=1nai−∑i=1nbi∑i=1n(ai−bi)=∑i=1nai−∑i=1nbi
 ∑i=1nai=∑i=1mai+∑i=m+1nai∑i=1nai=∑i=1mai+∑i=m+1nai
Sums and Powers of Integers ∑i=1ni=1+2+⋯+n=n(n+1)2∑i=1ni=1+2+⋯+n=n(n+1)2
 ∑i=1ni2=12+22+⋯+n2=n(n+1)(2n+1)6∑i=1ni2=12+22+⋯+n2=n(n+1)(2n+1)6
 ∑i=0ni3=13+23+⋯+n3=n2(n+1)24∑i=0ni3=13+23+⋯+n3=n2(n+1)24
Left-Endpoint Approximation A≈Ln=f(x0)Δx+f(x1)Δx+⋯+f(xn−1)Δx=∑i=1nf(xi−1)ΔxA≈Ln=f(x0)Δx+f(x1)Δx+⋯+f(xn−1)Δx=∑i=1nf(xi−1)Δx
Right-Endpoint Approximation A≈Rn=f(x1)Δx+f(x2)Δx+⋯+f(xn)Δx=∑i=1nf(xi)ΔxA≈Rn=f(x1)Δx+f(x2)Δx+⋯+f(xn)Δx=∑i=1nf(xi)Δx
Definite Integral ∫abf(x)dx=limn→∞∑i=1nf(xi*)Δx∫abf(x)dx=limn→∞∑i=1nf(xi*)Δx
Properties of the Definite Integral ∫aaf(x)dx=0∫aaf(x)dx=0
 ∫baf(x)dx=−∫abf(x)dx∫baf(x)dx=−∫abf(x)dx
 ∫ab[f(x)+g(x)]dx=∫abf(x)dx+∫abg(x)dx∫ab[f(x)+g(x)]dx=∫abf(x)dx+∫abg(x)dx
 ∫ab[f(x)−g(x)]dx=∫abf(x)dx−∫abg(x)dx∫ab[f(x)−g(x)]dx=∫abf(x)dx−∫abg(x)dx
 ∫abcf(x)dx=c∫abf(x)∫abcf(x)dx=c∫abf(x) for constant c
 ∫abf(x)dx=∫acf(x)dx+∫cbf(x)dx∫abf(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)=1b−a∫abf(x)dx.f(c)=1b−a∫abf(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))+C∫f[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)du∫abf(g(x))g'(x)dx=∫g(a)g(b)f(u)du
Integrals of Exponential Functions ∫exdx=ex+C∫exdx=ex+C
 ∫axdx=axlna+C∫axdx=axlna+C
Integration Formulas Involving Logarithmic Functions ∫x−1dx=ln|x|+C∫x−1dx=ln|x|+C
 ∫lnxdx=xlnx−x+C=x(lnx−1)+C∫lnxdx=xlnx−x+C=x(lnx−1)+C
 ∫logaxdx=xlna(lnx−1)+C∫logaxdx=xlna(lnx−1)+C
Integrals That Produce Inverse Trigonometric Functions ∫dua2−u2=sin−1(ua)+C∫dua2−u2=sin−1(ua)+C
 ∫dua2+u2=1atan−1(ua)+C∫dua2+u2=1atan−1(ua)+C
 ∫duuu2−a2=1asec−1(ua)+C∫duuu2−a2=1asec−1(ua)+C
PreviousNext
Order a print copy

As an Amazon Associate we earn from qualifying purchases.

Citation/Attribution

Want to cite, share, or modify this book? This book uses the Creative Commons Attribution-NonCommercial-ShareAlike License and you must attribute OpenStax.

Attribution information
  • If you are redistributing all or part of this book in a print format, then you must include on every physical page the following attribution:
    Access for free at https://openstax.org/books/calculus-volume-1/pages/1-introduction
  • If you are redistributing all or part of this book in a digital format, then you must include on every digital page view the following attribution:
    Access for free at https://openstax.org/books/calculus-volume-1/pages/1-introduction
Citation information

© Jan 13, 2023 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike License . The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.