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Calculus Volume 3

C | Review of Pre-Calculus

Calculus Volume 3C | Review of Pre-Calculus
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  1. Preface
  2. 1 Parametric Equations and Polar Coordinates
    1. Introduction
    2. 1.1 Parametric Equations
    3. 1.2 Calculus of Parametric Curves
    4. 1.3 Polar Coordinates
    5. 1.4 Area and Arc Length in Polar Coordinates
    6. 1.5 Conic Sections
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Chapter Review Exercises
  3. 2 Vectors in Space
    1. Introduction
    2. 2.1 Vectors in the Plane
    3. 2.2 Vectors in Three Dimensions
    4. 2.3 The Dot Product
    5. 2.4 The Cross Product
    6. 2.5 Equations of Lines and Planes in Space
    7. 2.6 Quadric Surfaces
    8. 2.7 Cylindrical and Spherical Coordinates
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Chapter Review Exercises
  4. 3 Vector-Valued Functions
    1. Introduction
    2. 3.1 Vector-Valued Functions and Space Curves
    3. 3.2 Calculus of Vector-Valued Functions
    4. 3.3 Arc Length and Curvature
    5. 3.4 Motion in Space
    6. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Chapter Review Exercises
  5. 4 Differentiation of Functions of Several Variables
    1. Introduction
    2. 4.1 Functions of Several Variables
    3. 4.2 Limits and Continuity
    4. 4.3 Partial Derivatives
    5. 4.4 Tangent Planes and Linear Approximations
    6. 4.5 The Chain Rule
    7. 4.6 Directional Derivatives and the Gradient
    8. 4.7 Maxima/Minima Problems
    9. 4.8 Lagrange Multipliers
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Chapter Review Exercises
  6. 5 Multiple Integration
    1. Introduction
    2. 5.1 Double Integrals over Rectangular Regions
    3. 5.2 Double Integrals over General Regions
    4. 5.3 Double Integrals in Polar Coordinates
    5. 5.4 Triple Integrals
    6. 5.5 Triple Integrals in Cylindrical and Spherical Coordinates
    7. 5.6 Calculating Centers of Mass and Moments of Inertia
    8. 5.7 Change of Variables in Multiple Integrals
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Chapter Review Exercises
  7. 6 Vector Calculus
    1. Introduction
    2. 6.1 Vector Fields
    3. 6.2 Line Integrals
    4. 6.3 Conservative Vector Fields
    5. 6.4 Green’s Theorem
    6. 6.5 Divergence and Curl
    7. 6.6 Surface Integrals
    8. 6.7 Stokes’ Theorem
    9. 6.8 The Divergence Theorem
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Chapter Review Exercises
  8. 7 Second-Order Differential Equations
    1. Introduction
    2. 7.1 Second-Order Linear Equations
    3. 7.2 Nonhomogeneous Linear Equations
    4. 7.3 Applications
    5. 7.4 Series Solutions of Differential Equations
    6. Key Terms
    7. Key Equations
    8. Key Concepts
    9. 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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