Schaum's Outline of Calculus

Cover
Title Page
Copyright Page
Contents
Preface
Chapter 1 Linear Coordinate Systems. Absolute Value. Inequalities.
Linear Coordinate System
Finite Intervals
Infinite Intervals
Inequalities
Solved Problems
Supplementary Problems
Chapter 2 Rectangular Coordinate Systems
Coordinate Axes
Coordinates
Quadrants
The Distance Formula
The Midpoint Formulas
Proofs of Geometric Theorems
Solved Problems
Supplementary Problems
Chapter 3 Lines
The Steepness of a Line
The Sign of the Slope
Slope and Steepness
Equations of Lines
A Point–Slope Equation
Slope–Intercept Equation
Parallel Lines
Perpendicular Lines
Solved Problems
Supplementary Problems
Chapter 4 Circles
Equations of Circles
The Standard Equation of a Circle
Solved Problems
Supplementary Problems
Chapter 5 Equations and Their Graphs
The Graph of an Equation
Parabolas
Ellipses
Hyperbolas
Conic Sections
Solved Problems
Supplementary Problems
Chapter 6 Functions
Solved Problems
Supplementary Problems
Chapter 7 Limits
Limit of a Function
Right and Left Limits
Theorems on Limits
Infinity
Solved Problems
Supplementary Problems
Chapter 8 Continuity
Continuous Function
Solved Problems
Supplementary Problems
Chapter 9 The Derivative
Delta Notation
The Derivative
Notation for Derivatives
Differentiability
Solved Problems
Supplementary Problems
Chapter 10 Rules for Differentiating Functions
Differentiation
Composite Functions. The Chain Rule.
Chain Rule
Alternative Formulation of the Chain Rule
Inverse Functions
Notation
Higher Derivatives
Notation
Solved Problems
Supplementary Problems
Chapter 11 Implicit Differentiation
Implicit Functions
Derivatives of Higher Order
Solved Problems
Supplementary Problems
Chapter 12 Tangent and Normal Lines
The Angles of Intersection
Solved Problems
Supplementary Problems
Chapter 13 Law of the Mean. Increasing and Decreasing Functions.
Relative Maximum and Minimum
Increasing and Decreasing Functions
Solved Problems
Supplementary Problems
Chapter 14 Maximum and Minimum Values
Critical Numbers
Second Derivative Test for Relative Extrema
First Derivative Test
Case {+, -}
Case {-, +}
Cases {+, +} and {-, -}
Absolute Maximum and Minimum
Tabular Method for Finding the Absolute Maximum and Minimum
Solved Problems
Supplementary Problems
Chapter 15 Curve Sketching. Concavity. Symmetry.
Concavity
Points of Inflection
Vertical Asymptotes
Horizontal Asymptotes
Symmetry
Inverse Functions and Symmetry
Even and Odd Functions
Hints for Sketching the Graph G of y = f(x)
Solved Problems
Supplementary Problems
Chapter 16 Review of Trigonometry
Angle Measure
Directed Angles
Sine and Cosine Functions
Solved Problems
Supplementary Problems
Chapter 17 Differentiation of Trigonometric Functions
Continuity of cos x and sin x
Graph of sin x
Graph of cos x
Other Trigonometric Functions
Derivatives
Other Relationships
Graph of y = tan x
Graph of y = sec x
Angles Between Curves
Solved Problems
Supplementary Problems
Chapter 18 Inverse Trigonometric Functions
The Derivative of sin-1 x
The Inverse Cosine Function
The Inverse Tangent Function
Solved Problems
Supplementary Problems
Chapter 19 Rectilinear and Circular Motion
Rectilinear Motion
Motion Under the Influence of Gravity
Circular Motion
Solved Problems
Supplementary Problems
Chapter 20 Related Rates
Solved Problems
Supplementary Problems
Chapter 21 Differentials. Newton’s Method.
The Differential
Definition
Newton’s Method
Solved Problems
Supplementary Problems
Chapter 22 Antiderivatives
Laws for Antiderivatives
Solved Problems
Supplementary Problems
Chapter 23 The Definite Integral. Area Under a Curve.
Sigma Notation
Area Under a Curve
Properties of the Definite Integral
Solved Problems
Supplementary Problems
Chapter 24 The Fundamental Theorem of Calculus
Mean Value Theorem for Integrals
Average Value of a Function on a Closed Interval
Fundamental Theorem of Calculus
Change of Variable in a Definite Integral
Solved Problems
Supplementary Problems
Chapter 25 The Natural Logarithm
The Natural Logarithm
Definition
Properties of the Natural Logarithm
Solved Problems
Supplementary Problems
Chapter 26 Exponential and Logarithmic Functions
Definition
Properties of ex
Definition
The General Exponential Function
Definition
General Logarithmic Functions
Definition
Solved Problems
Supplementary Problems
Chapter 27 L’Hôpital’s Rule
L’Hôpital’s Rule
Indeterminate Type 0 • ∞
Indeterminate Type ∞ - ∞
Indeterminate Types 00, ∞0, and 1∞
Solved Problems
Supplementary Problems
Chapter 28 Exponential Growth and Decay
Half-Life
Solved Problems
Supplementary Problems
Chapter 29 Applications of Integration I: Area and Arc Length
Area Between a Curve and the Y-Axis
Areas Between Curves
Arc Length
Solved Problems
Supplementary Problems
Chapter 30 Applications of Integration II: Volume
Disk Formula
Washer Method
Cylindrical Shell Method
Difference of Shells Formula
Cross-Section Formula (Slicing Formula)
Solved Problems
Supplementary Problems
Chapter 31 Techniques of Integration I: Integration by Parts
Solved Problems
Supplementary Problems
Chapter 32 Techniques of Integration II: Trigonometric Integrands and Trigonometric Substitutions
Trigonometric Integrands
Trigonometric Substitutions
Solved Problems
Supplementary Problems
Chapter 33 Techniques of Integration III: Integration by Partial Fractions
Method of Partial Fractions
Case I
Case II
Case III
Case IV
Solved Problems
Supplementary Problems
Chapter 34 Techniques of Integration IV: Miscellaneous Substitutions
Solved Problems
Supplementary Problems
Chapter 35 Improper Integrals
Infinite Limits of Integration
Discontinuities of the Integrand
Solved Problems
Supplementary Problems
Chapter 36 Applications of Integration III: Area of a Surface of Revolution
Solved Problems
Supplementary Problems
Chapter 37 Parametric Representation of Curves
Parametric Equations
(37.1) First Derivative
(37.2) Second Derivative
Arc Length for a Parametric Curve
Solved Problems
Supplementary Problems
Chapter 38 Curvature
Derivative of Arc Length
Curvature
The Radius of Curvature
The Circle of Curvature
The Center of Curvature
The Evolute
Solved Problems
Supplementary Problems
Chapter 39 Plane Vectors
Scalars and Vectors
Sum and Difference of Two Vectors
Components of a Vector
Scalar Product (or Dot Product)
Scalar and Vector Projections
Differentiation of Vector Functions
Solved Problems
Supplementary Problems
Chapter 40 Curvilinear Motion
Velocity in Curvilinear Motion
Acceleration in Curvilinear Motion
Tangential and Normal Components of Acceleration
Solved Problems
Supplementary Problems
Chapter 41 Polar Coordinates
Polar and Rectangular Coordinates
Some Typical Polar Curves
Angle of Inclination
Points of Intersection
Angle of Intersection
The Derivative of the Arc Length
Curvature
Solved Problems
Supplementary Problems
Chapter 42 Infinite Sequences
Infinite Sequences
Limit of a Sequence
Monotonic Sequences
Solved Problems
Supplementary Problems
Chapter 43 Infinite Series
Geometric Series
Solved Problems
Supplementary Problems
Chapter 44 Series with Positive Terms. The Integral Test. Comparison Tests.
Series of Positive Terms
Solved Problems
Supplementary Problems
Chapter 45 Alternating Series. Absolute and Conditional Convergence. The Ratio Test.
Alternating Series
Definition
Solved Problems
Supplementary Problems
Chapter 46 Power Series
Power Series
Uniform Convergence
Solved Problems
Supplementary Problems
Chapter 47 Taylor and Maclaurin Series. Taylor’s Formula with Remainder.
Taylor and Maclaurin Series
Applications of Taylor’s Formula with Remainder
Solved Problems
Supplementary Problems
Chapter 48 Partial Derivatives
Functions of Several Variables
Limits
Continuity
Partial Derivatives
Partial Derivatives of Higher Order
Solved Problems
Supplementary Problems
Chapter 49 Total Differential. Differentiability. Chain Rules.
Total Differential
Differentiability
Chain Rules
Chain Rule (2 ã 1)
Chain Rule (2 ã 2)
Implicit Differentiation
Solved Problems
Supplementary Problems
Chapter 50 Space Vectors
Vectors in Space
Direction Cosines of a Vector
Determinants
Vector Perpendicular to Two Vectors
Vector Product of Two Vectors
Triple Scalar Product
Triple Vector Product
The Straight Line
The Plane
Solved Problems
Supplementary Problems
Chapter 51 Surfaces and Curves in Space
Planes
Spheres
Cylindrical Surfaces
Ellipsoid
Elliptic Paraboloid
Elliptic Cone
Hyperbolic Paraboloid
Hyperboloid of One Sheet
Hyperboloid of Two Sheets
Tangent Line and Normal Plane to a Space Curve
Tangent Plane and Normal Line to a Surface
Surface of Revolution
Solved Problems
Supplementary Problems
Chapter 52 Directional Derivatives. Maximum and Minimum Values.
Directional Derivatives
Relative Maximum and Minimum Values
Absolute Maximum and Minimum Values
Solved Problems
Supplementary Problems
Chapter 53 Vector Differentiation and Integration
Vector Differentiation
Space Curves
Surfaces
The Operation ∇
Divergence and Curl
Integration
Line Integrals
Solved Problems
Supplementary Problems
Chapter 54 Double and Iterated Integrals
The Double Integral
The Iterated Integral
Solved Problems
Supplementary Problems
Chapter 55 Centroids and Moments of Inertia of Plane Areas
Plane Area by Double Integration
Centroids
Moments of Inertia
Solved Problems
Supplementary Problems
Chapter 56 Double Integration Applied to Volume Under a Surface and the Area of a Curved Surface
Solved Problems
Supplementary Problems
Chapter 57 Triple Integrals
Cylindrical and Spherical Coordinates
The Triple Integral
Evaluation of Triple Integrals
Centroids and Moments of Inertia
Solved Problems
Supplementary Problems
Chapter 58 Masses of Variable Density
Solved Problems
Supplementary Problems
Chapter 59 Differential Equations of First and Second Order
Separable Differential Equations
Homogeneous Functions
Integrating Factors
Second-Order Equations
Solved Problems
Supplementary Problems
Appendix A Trigonometric Formulas
Appendix B Geometric Formulas
Index