# Calculus And Vectors 12: Study Guide And University Handbook

## byChris Knowles, Antonietta Lenjosek

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• Extend practice and exercises, and build skills needed for university bound students
• Includes key concepts with expanded examples
• Consumable format which can serve as a reference tool for students as they enter university
Title:Calculus And Vectors 12: Study Guide And University HandbookFormat:PaperbackDimensions:10.8 × 8.4 × 0.8 inPublished:August 25, 2008Publisher:McGraw Hill School IndigeLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:0070735891

ISBN - 13:9780070735897

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## Reviews

Overview
Formulas

Chapter 1 Rates of Change
1.1 Rates of Change and the Slope of a Curve
Average Rate of Change
Instantaneous Rate of Change
1.2 Rates of Change Using Equations
Difference Quotient
Estimate the Instantaneous Rate of Change
Estimate the Slope of a Tangent
1.3 Limits
Limit of a Sequence
Limit of a Function
One-sided Limits
1.4 Limits and Continuity
Evaluate Limits Algebraically
Limit Properties
Continuous and Discontinuous Functions
Limits Involving Asymptotes
Indeterminate Forms
1.5 Introduction to Derivatives
First Principles Definition
Leibniz Notation
Differentiate Rational Functions
Non-Differentiable Functions
Solve Rate and Tangent Problems Using First Principles
Challenge Questions
Chapter 1 Checklist

Chapter 2 Derivatives
2.1 Derivative of a Polynomial Function
Derivative Rules: The Constant Rule, The Power Rule, The Sum and Difference Rules, The Constant Multiple Rule
Rational Exponents and the Power Rule
Differentiate Powers with Negative Exponents
Applications of Polynomial Derivatives
2.2 The Product Rule
2.3 Velocity, Acceleration, and Second Derivatives
Relationship Between the First and Second Derivative
Determining the Second Derivative
Relationship Between Displacement, Velocity, and Acceleration
2.4 The Chain Rule
Differentiate Composite Functions
Leibniz Form of the Chain Rule
Power of a Function Rule
Combining Derivative Rules and the Chain Rule
2.5 Derivatives of Quotients
Differentiating a Simple Quotient Function
The Quotient Rule
2.6 Rate of Change Problems
Functions Pertaining to Business: Demand, Revenue, Cost, and Profit Functions
Derivatives of Business Functions: Marginal Cost, Marginal Revenue, and Marginal Profit
Applications of Derivatives in Physical Sciences: Kinetic Energy, Electrical Currents, Linear Density
Challenge Questions
Chapter 2 Checklist

Chapter 3 Curve Sketching
3.1 Increasing and Decreasing Functions
Intervals of Increase and Decrease
Sketch Functions Using the First Derivative
3.2 Maxima and Minima
Critical Values
Local Maximum and Minimum Values
Absolute Maximum and Minimum Values
3.3 Concavity and the Second Derivative Test
Second Derivative Test
Point of Inflection
Intervals of Concavity
3.4 Simple Rational Functions
Vertical Asymptotes
Derivatives of Rational Functions
Concavity of Rational Functions
3.5 Putting It All Together
Steps to Analyse a Function
Analyse and Sketch Functions
3.6 Optimization Problems
Area, Surface Area, Volume Problems
Cost, Revenue Problems
Challenge Questions
Chapter 3 Checklist

Chapter 4 Derivatives of Sinusoidal Functions
4.1 Instantaneous Rates of Change of the Sinusoidal Functions
Derivative of a Sinusoidal Function
4.2 Derivatives of the Sine and Cosine Functions
Constant Multiple Rule
Sum and Difference Rules
Slope at a Point
Equation of a Tangent Line
4.3 Differentiation Rules for Sinusoidal Functions
Chain Rule
Power of a Function Rule
Product Rule
Combining Derivative Rules
4.4 Applications of Sinusoidal Functions and Their Derivatives
Models of Periodic Behaviour
Challenge Questions
Chapter 4 Checklist

Chapter 5 Exponential and Logarithmic Functions
5.1 Rate of Change and the Number e
Nature of the Rate of Change
Value of Number e
5.2 The Natural Logarithm
Value of Number e
Natural Logarithm
Applications
5.3 Derivatives of Exponential Functions
Derivative of f(x) = bx
Equation of a Tangent Line
Applications
5.4 Differentiation Rules for Exponential Functions
Product Rule
Chain Rule
Difference Rule
Combining Rules
Extreme Values
Applications
5.5 Making Connections: Exponential Models
Modelling Using Exponential Functions and their Derivatives
Representations of Exponential Models
Challenge Questions
Chapter 5 Checklist

Chapter 6 Geometric Vectors
6.1 Introduction to Vectors
Vectors and Scalars
True Bearings
Equivalent and Opposite Vectors
6.2 Addition and Subtraction of Vectors
Parallel Vectors
Opposite Vectors
The Zero Vector
Properties of Vector Addition and Subtraction
6.3 Multiplying a Vector by a Scalar
Scalar Multiplication
Distributive Property
Collinear Vectors
Vector Properties for Scalar Multiplication
Linear Combinations of Vectors
Rectangular Vector Components
Resultant Vector
Equilibrant Vector
Applications Involving Velocities and Forces
6.5 Resolution of Vectors into Rectangular Components
Horizontal and Vertical Components of a Force
Challenge Questions
Chapter 6 Checklist

Chapter 7 Cartesian Vectors
7.1 Cartesian Vectors
Position Vector
Unit Vectors
Magnitude of a Vector
Operations With Cartesian Vectors
Cartesian Vectors Between Two Points
Forces and Velocities as Cartesian Vectors
7.2 Dot Product
Work
Properties of the Dot Product
Calculate Dot Products of Vectors
7.3 Applications of the Dot Product
Work in Cartesian Form
The Angle Between Two Cartesian Vectors
Vector Projections
7.4 Vectors in Three-Space
Octants in a 3-D Graph
Plot Points in 3-D
3-D Cartesian Vectors
Magnitude of a Cartesian Vector
Operations with Cartesian Vectors in 3-D
Collinear Vectors
Orthogonal Vectors
Properties of Cartesian Vector Operations in Three-Space
7.5 The Cross Product and Its Properties
Cross Product in Cartesian Form
Properties of the Cross Product of Cartesian Vectors
7.6 Applications of the Dot Product and Cross Product
Torque
Vector Projections and Work in Three-Space
Triple Scalar Product
Volume of a Parallelepiped
Challenge Questions
Chapter 7 Checklist

Chapter 8 Lines and Planes
8.1 Equations of Lines in Two-Space and Three-Space
Vector Equation of a Line in Two-Space
Parametric Equations of a Line in Two-Space
Vector Equations of Lines in Three-Space
Parametric Equations of Lines in Three-Space
8.2 Equations of Planes
Vector Equations of Planes in Three-Space
Parametric Equations of Planes in Three-Space
Scalar Equations of Planes in Three-Space
8.3 Properties of Planes
Scalar Equations of Planes in Three-Space
8.4 Intersections of Lines in Two-Space and Three-Space
Linear Systems in Two-Space
Linear Systems in Three-Space
Intersection of Lines
Distance Between Two Skew Lines
8.5 Intersections of Lines and Planes
Intersection of a Line and a Plane
Distance From a Point to a Plane
8.6 Intersection of Planes
Intersection of Two Planes
Consistent and Inconsistent Systems of Three Planes
Solving Systems With Three Planes
Analysing Inconsistent Solutions
Solving Systems of Equations Using Matrices
Elementary Row Operations
Row Reduced Echelon Form
Solve Dependent or Inconsistent Systems
Challenge Questions
Chapter 8 Checklist

University Preparation

CALCULUS
Implicit Differentiation
Derivatives of Logarithmic Differentiation
Related Rates
Antiderivatives
Integration: The Substitution Rule and Integration by Parts

VECTORS
Solving Systems of Equations
Practice Exam