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Calculus Volume 2

C | Review of Pre-Calculus

Calculus Volume 2C | Review of Pre-Calculus
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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

Formulas from Geometry

A=area,A=area, V=Volume,andV=Volume,and S=lateral surface areaS=lateral surface area

The figure shows five geometric figures. The first is a parallelogram with height labeled as h and base as b. Below the figure is the formula for area, A = bh. The second is a triangle with height labeled as h and base as b. Below the figure is the formula for area, A = (1/2)bh.. The third is a trapezoid with the top horizontal side labeled as a, height as h, and base as b. Below the figure is the formula for area, A = (1/2)(a + b)h. The fourth is a circle with radius labeled as r. Below the figure is the formula for area, A= (pi)(r^2), and the formula for circumference, C = 2(pi)r. The fifth is a sector of a circle with radius labeled as r, sector length as s, and angle as theta. Below the figure is the formula for area, A = (1/2)r^2(theta), and sector length, s = r(theta) (theta in radians). The figure shows three solid figures. The first is a cylinder with height labeled as h and radius as r. Below the figure are the formulas for volume, V = (pi)(r^2)h, and surface area, S = 2(pi)rh. The second is a cone with height labeled as h, radius as r, and lateral side length as l. Below the figure are the formulas for volume, V = (1/3)(pi)(r^2)h, and surface area, S = (pi)rl. The third is a sphere with radius labeled as r. Below the figure are the formulas for volume, V = (4/3)(pi)(r^3), and surface area, S = 4(pi)r^2.

Formulas from Algebra

Laws of Exponents

xmxn=xm+nxmxn=xmn(xm)n=xmn xn=1xn(xy)n=xnyn(xy)n=xnyn x1/n=xnxyn=xnynxyn=xnyn xm/n=xmn=(xn)mxmxn=xm+nxmxn=xmn(xm)n=xmn xn=1xn(xy)n=xnyn(xy)n=xnyn x1/n=xnxyn=xnynxyn=xnyn xm/n=xmn=(xn)m

Special Factorizations

x2y2=(x+y)(xy)x3+y3=(x+y)(x2xy+y2)x3y3=(xy)(x2+xy+y2)x2y2=(x+y)(xy)x3+y3=(x+y)(x2xy+y2)x3y3=(xy)(x2+xy+y2)

Quadratic Formula

If ax2+bx+c=0,ax2+bx+c=0, then x=b±b24ca2a.x=b±b24ca2a.

Binomial Theorem

(a+b)n=an+(n1)an1b+(n2)an2b2++(nn1)abn1+bn,(a+b)n=an+(n1)an1b+(n2)an2b2++(nn1)abn1+bn,

where (nk)=n(n1)(n2)(nk+1)k(k1)(k2)321=n!k!(nk)!(nk)=n(n1)(n2)(nk+1)k(k1)(k2)321=n!k!(nk)!

Formulas from Trigonometry

Right-Angle Trigonometry

sinθ=opphypcscθ=hypoppcosθ=adjhypsecθ=hypadjtanθ=oppadjcotθ=adjoppsinθ=opphypcscθ=hypoppcosθ=adjhypsecθ=hypadjtanθ=oppadjcotθ=adjopp

The figure shows a right triangle with the longest side labeled hyp, the shorter leg labeled as opp, and the longer leg labeled as adj. The angle between the hypotenuse and the adjacent side is labeled theta.

Trigonometric Functions of Important Angles

θθ RadiansRadians sinθsinθ cosθcosθ tanθtanθ
0°0° 00 00 11 00
30°30° π/6π/6 1/21/2 3/23/2 3/33/3
45°45° π/4π/4 2/22/2 2/22/2 11
60°60° π/3π/3 3/23/2 1/21/2 33
90°90° π/2π/2 11 00

Fundamental Identities

sin2θ+cos2θ=1sin(θ)=sinθ 1+tan2θ=sec2θcos(θ)=cosθ1+cot2θ=csc2θtan(θ)=tanθsin(π2θ)=cosθsin(θ+2π)=sinθ cos(π2θ)=sinθcos(θ+2π)=cosθ tan(π2θ)=cotθtan(θ+π)=tanθsin2θ+cos2θ=1sin(θ)=sinθ 1+tan2θ=sec2θcos(θ)=cosθ1+cot2θ=csc2θtan(θ)=tanθsin(π2θ)=cosθsin(θ+2π)=sinθ cos(π2θ)=sinθcos(θ+2π)=cosθ tan(π2θ)=cotθtan(θ+π)=tanθ

Law of Sines

sinAa=sinBb=sinCcsinAa=sinBb=sinCc

The figure shows a nonright triangle with vertices labeled A, B, and C. The side opposite angle A is labeled a. The side opposite angle B is labeled b. The side opposite angle C is labeled c.

Law of Cosines

a2=b2+c22bccosAb2=a2+c22accosBc2=a2+b22abcosCa2=b2+c22bccosAb2=a2+c22accosBc2=a2+b22abcosC

Addition and Subtraction Formulas

sin(x+y)=sinxcosy+cosxsinysin(xy)=sinxcosycosxsinycos(x+y)=cosxcosysinxsinycos(xy)=cosxcosy+sinxsinytan(x+y)=tanx+tany1tanxtanytan(xy)=tanxtany1+tanxtanysin(x+y)=sinxcosy+cosxsinysin(xy)=sinxcosycosxsinycos(x+y)=cosxcosysinxsinycos(xy)=cosxcosy+sinxsinytan(x+y)=tanx+tany1tanxtanytan(xy)=tanxtany1+tanxtany

Double-Angle Formulas

sin2x=2sinxcosxcos2x=cos2xsin2x=2cos2x1=12sin2xtan2x=2tanx1tan2xsin2x=2sinxcosxcos2x=cos2xsin2x=2cos2x1=12sin2xtan2x=2tanx1tan2x

Half-Angle Formulas

sin2x=1cos2x2cos2x=1+cos2x2sin2x=1cos2x2cos2x=1+cos2x2

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