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# Calculus And Vectors 12: Study Guide And University Handbook

## byChris Knowles, Antonietta Lenjosek

### Paperback | August 25, 2008

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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

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The following ISBNs are associated with this title:

ISBN - 10:0070735891

ISBN - 13:9780070735897

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Table of Contents

**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

Quadrant Bearings

Equivalent and Opposite Vectors

6.2 Addition and Subtraction of Vectors

Parallel Vectors

Opposite Vectors

The Zero Vector

Parallelogram Method of Adding Vectors

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

6.4 Applications of Vector Addition

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

Answers