Calculus Lifesaver All the Tools You Need to Excel at Calculus

ISBN-10: 0691130884

ISBN-13: 9780691130880

Edition: 2007

Authors: Adrian Banner

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For many students, calculus can be the most mystifying and frustrating course they will ever take.The Calculus Lifesaverprovides students with the essential tools they need not only to learn calculus, but to excel at it. All of the material in this user-friendly study guide has been proven to get results. The book arose from Adrian Banner's popular calculus review course at Princeton University, which he developed especially for students who are motivated to earn A's but get only average grades on exams. The complete course will be available for free on the Web in a series of videotaped lectures. This study guide works as a supplement to any single-variable calculus course or textbook. Coupled with a selection of exercises, the book can also be used as a textbook in its own right. The style is informal, non-intimidating, and even entertaining, without sacrificing comprehensiveness. The author elaborates standard course material with scores of detailed examples that treat the reader to an "inner monologue"--the train of thought students should be following in order to solve the problem--providing the necessary reasoning as well as the solution. The book's emphasis is on building problem-solving skills. Examples range from easy to difficult and illustrate the in-depth presentation of theory. The Calculus Lifesavercombines ease of use and readability with the depth of content and mathematical rigor of the best calculus textbooks. It is an indispensable volume for any student seeking to master calculus. Serves as a companion to any single-variable calculus textbook Informal, entertaining, and not intimidating Informative videos that follow the book--a full forty-eight hours of Banner's Princeton calculus-review course--will be available through the book's Web site at More than 475 examples (ranging from easy to hard) provide step-by-step reasoning Theorems and methods justified and connections made to actual practice Difficult topics such as improper integrals and infinite series covered in detail Tried and tested by students taking freshman calculus
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Book details

List price: $24.95
Copyright year: 2007
Publisher: Princeton University Press
Publication date: 3/25/2007
Binding: Paperback
Pages: 752
Size: 7.50" wide x 10.25" long x 1.75" tall
Weight: 3.212
Language: English

How to Use This Book to Study for an Exam
Two all-purpose study tips
Key sections for exam review (by topic)
Functions, Graphs, and Lines
Interval notation
Finding the domain
Finding the range using the graph
The vertical line test
Inverse Functions
The horizontal line test
Finding the inverse
Restricting the domain
Inverses of inverse functions
Composition of Functions
Odd and Even Functions
Graphs of Linear Functions
Common Functions and Graphs
Review of Trigonometry
The Basics
Extending the Domain of Trig Functions
The ASTC method
Trig functions outside [0,2[pi]]
The Graphs of Trig Functions
Trig Identities
Introduction to Limits
Limits: The Basic Idea
Left-Hand and Right-Hand Limits
When the Limit Does Not Exist
Limits at [infinity] and [infinity]
Large numbers and small numbers
Two Common Misconceptions about Asymptotes
The Sandwich Principle
Summary of Basic Types of Limits
How to Solve Limit Problems Involving Polynomials
Limits Involving Rational Functions as x [RightArrow] a
Limits Involving Square Roots as x [RightArrow] a
Limits Involving Rational Functions as x [RightArrow infinity] a
Method and examples
Limits Involving Poly-type Functions as x [RightArrow infinity]
Limits Involving Rational Functions as x [RightArrow infinity]
Limits Involving Absolute Values
Continuity and Differentiability
Continuity at a point
Continuity on an interval
Examples of continuous functions
The Intermediate Value Theorem
A harder IVT example
Maxima and minima of continuous functions
Average speed
Displacement and velocity
Instantaneous velocity
The graphical interpretation of velocity
Tangent lines
The derivative function
The derivative as a limiting ratio
The derivative of linear functions
Second and higher-order derivatives
When the derivative does not exist
Differentiability and continuity
How to Solve Differentiation Problems
Finding Derivatives Using the Definition
Finding Derivatives (the Nice Way)
Constant multiples of functions
Sums and differences of functions
Products of functions via the product rule
Quotients of functions via the quotient rule
Composition of functions via the chain rule
A nasty example
Justification of the product rule and the chain rule
Finding the Equation of a Tangent Line
Velocity and Acceleration
Constant negative acceleration
Limits Which Are Derivatives in Disguise
Derivatives of Piecewise-Defined Functions
Sketching Derivative Graphs Directly
Trig Limits and Derivatives
Limits Involving Trig Functions
The small case
Solving problems-the small case
The large case
The "other" case
Proof of an important limit
Derivatives Involving Trig Functions
Examples of differentiating trig functions
Simple harmonic motion
A curious function
Implicit Differentiation and Related Rates
Implicit Differentiation
Techniques and examples
Finding the second derivative implicitly
Related Rates
A simple example
A slightly harder example
A much harder example
A really hard example
Exponentials and Logarithms
The Basics
Review of exponentials
Review of logarithms
Logarithms, exponentials, and inverses
Log rules
Definition of e
A question about compound interest
The answer to our question
More about e and logs
Differentiation of Logs and Exponentials
Examples of differentiating exponentials and logs
How to Solve Limit Problems Involving Exponentials or Logs
Limits involving the definition of e
Behavior of exponentials near 0
Behavior of logarithms near 1
Behavior of exponentials near [infinity] or -[infinity]
Behavior of logs near [infinity]
Behavior of logs near 0
Logarithmic Differentiation
The derivative of x[superscript a]
Exponential Growth and Decay
Exponential growth
Exponential decay
Hyperbolic Functions
Inverse Functions and Inverse Trig Functions
The Derivative and Inverse Functions
Using the derivative to show that an inverse exists
Derivatives and inverse functions: what can go wrong
Finding the derivative of an inverse function
A big example
Inverse Trig Functions
Inverse sine
Inverse cosine
Inverse tangent
Inverse secant
Inverse cosecant and inverse cotangent
Computing inverse trig functions
Inverse Hyperbolic Functions
The rest of the inverse hyperbolic functions
The Derivative and Graphs
Extrema of Functions
Global and local extrema
The Extreme Value Theorem
How to find global maxima and minima
Rolle's Theorem
The Mean Value Theorem
Consequences of the Mean Value Theorem
The Second Derivative and Graphs
More about points of inflection
Classifying Points Where the Derivative Vanishes
Using the first derivative
Using the second derivative
Sketching Graphs
How to Construct a Table of Signs
Making a table of signs for the derivative
Making a table of signs for the second derivative
The Big Method
An example without using derivatives
The full method: example 1
The full method: example 2
The full method: example 3
The full method: example 4
Optimization and Linearization
An easy optimization example
Optimization problems: the general method
An optimization example
Another optimization example
Using implicit differentiation in optimization
A difficult optimization example
Linearization in general
The differential
Linearization summary and examples
The error in our approximation
Newton's Method
L'Hopital's Rule and Overview of Limits
L'Hopital's Rule
Type A: 0/0 case
Type A: [PlusMinus infinity] / [PlusMinus infinity] case
Type B1 ([infinity] - [infinity])
Type B2 (0 x [PlusMinus infinity])
Type C (1[PlusMinus infinity], 0[superscript 0], or [infinity superscript 0])
Summary of l'Hopital's Rule types
Overview of Limits
Introduction to Integration
Sigma Notation
A nice sum
Telescoping series
Displacement and Area
Three simple cases
A more general journey
Signed area
Continuous velocity
Two special approximations
Definite Integrals
The Basic Idea
Some easy examples
Definition of the Definite Integral
An example of using the definition
Properties of Definite Integrals
Finding Areas
Finding the unsigned area
Finding the area between two curves
Finding the area between a curve and the y-axis
Estimating Integrals
A simple type of estimation
Averages and the Mean Value Theorem for Integrals
The Mean Value Theorem for integrals
A Nonintegrable Function
The Fundamental Theorems of Calculus
Functions Based on Integrals of Other Functions
The First Fundamental Theorem
Introduction to antiderivatives
The Second Fundamental Theorem
Indefinite Integrals
How to Solve Problems: The First Fundamental Theorem
Variation 1: variable left-hand limit of integration
Variation 2: one tricky limit of integration
Variation 3: two tricky limits of integration
Variation 4: limit is a derivative in disguise
How to Solve Problems: The Second Fundamental Theorem
Finding indefinite integrals
Finding definite integrals
Unsigned areas and absolute values
A Technical Point
Proof of the First Fundamental Theorem
Techniques of Integration, Part One
Substitution and definite integrals
How to decide what to substitute
Theoretical justification of the substitution method
Integration by Parts
Some variations
Partial Fractions
The algebra of partial fractions
Integrating the pieces
The method and a big example
Techniques of Integration, Part Two
Integrals Involving Trig Identities
Integrals Involving Powers of Trig Functions
Powers of sin and/or cos
Powers of tan
Powers of sec
Powers of cot
Powers of csc
Reduction formulas
Integrals Involving Trig Substitutions
Type 1: [Characters not reproducible]
Type 2: [Characters not reproducible]
Type 3: [Characters not reproducible]
Completing the square and trig substitutions
Summary of trig substitutions
Technicalities of square roots and trig substitutions
Overview of Techniques of Integration
Improper Integrals: Basic Concepts
Convergence and Divergence
Some examples of improper integrals
Other blow-up points
Integrals over Unbounded Regions
The Comparison Test (Theory)
The Limit Comparison Test (Theory)
Functions asymptotic to each other
The statement of the test
The p-test (Theory)
The Absolute Convergence Test
Improper Integrals: How to Solve Problems
How to Get Started
Splitting up the integral
How to deal with negative function values
Summary of Integral Tests
Behavior of Common Functions near [infinity] and -[infinity]
Polynomials and poly-type functions near [infinity] and -[infinity]
Trig functions near [infinity] and -[infinity]
Exponentials near [infinity] and -[infinity]
Logarithms near [infinity]
Behavior of Common Functions near 0
Polynomials and poly-type functions near 0
Trig functions near 0
Exponentials near 0
Logarithms near 0
The behavior of more general functions near 0
How to Deal with Problem Spots Not at 0 or [infinity]
Sequences and Series: Basic Concepts
Convergence and Divergence of Sequences
The connection between sequences and functions
Two important sequences
Convergence and Divergence of Series
Geometric series (theory)
The nth Terra Test (Theory)
Properties of Both Infinite Series and Improper Integrals
The comparison test (theory)
The limit comparison test (theory)
The p-test (theory)
The absolute convergence test
New Tests for Series
The ratio test (theory)
The root test (theory)
The integral test (theory)
The alternating series test (theory)
How to Solve Series Problems
How to Evaluate Geometric Series
How to Use the nth Term Test
How to Use the Ratio Test
How to Use the Root Test
How to Use the Integral Test
Comparison Test, Limit Comparison Test, and p-test
How to Deal with Series with Negative Terms
Taylor Polynomials, Taylor Series, and Power Series
Approximations and Taylor Polynomials
Linearization revisited
Quadratic approximations
Higher-degree approximations
Taylor's Theorem
Power Series and Taylor Series
Power series in general
Taylor series and Maclaurin series
Convergence of Taylor series
A Useful Limit
How to Solve Estimation Problems
Summary of Taylor Polynomials and Series
Finding Taylor Polynomials and Series
Estimation Problems Using the Error Term
First example
Second example
Third example
Fourth example
Fifth example
General techniques for estimating the error term
Another Technique for Estimating the Error
Taylor and Power Series: How to Solve Problems
Convergence of Power Series
Radius of convergence
How to find the radius and region of convergence
Getting New Taylor Series from Old Ones
Substitution and Taylor series
Differentiating Taylor series
Integrating Taylor series
Adding and subtracting Taylor series
Multiplying Taylor series
Dividing Taylor series
Using Power and Taylor Series to Find Derivatives
Using Maclaurin Series to Find Limits
Parametric Equations and Polar Coordinates
Parametric Equations
Derivatives of parametric equations
Polar Coordinates
Converting to and from polar coordinates
Sketching curves in polar coordinates
Finding tangents to polar curves
Finding areas enclosed by polar curves
Complex Numbers
The Basics
Complex exponentials
The Complex Plane
Converting to and from polar form
Taking Large Powers of Complex Numbers
Solving z[superscript n] = w
Some variations
Solving e[superscript z] = w
Some Trigonometric Series
Euler's Identity and Power Series
Volumes, Arc Lengths, and Surface Areas
Volumes of Solids of Revolution
The disc method
The shell method
Summary...and variations
Variation 1: regions between a curve and the y-axis
Variation 2: regions between two curves
Variation 3: axes parallel to the coordinate axes
Volumes of General Solids
Arc Lengths
Parametrization and speed
Surface Areas of Solids of Revolution
Differential Equations
Introduction to Differential Equations
Separable First-order Differential Equations
First-order Linear Equations
Why the integrating factor works
Constant-coefficient Differential Equations
Solving first-order homogeneous equations
Solving second-order homogeneous equations
Why the characteristic quadratic method works
Nonhomogeneous equations and particular solutions
Finding a particular solution
Examples of finding particular solutions
Resolving conflicts between y[subscript P] and y[subscript H]
Initial value problems (constant-coefficient linear)
Modeling Using Differential Equations
Limits and Proofs
Formal Definition of a Limit
A little game
The actual definition
Examples of using the definition
Making New Limits from Old Ones
Sum and differences of limits-proofs
Products of limits-proof
Quotients of limits-proof
The sandwich principle-proof
Other Varieties of Limits
Infinite limits
Left-hand and right-hand limits
Limits at [infinity] and -[infinity]
Two examples involving trig
Continuity and Limits
Composition of continuous functions
Proof of the Intermediate Value Theorem
Proof of the Max-Min Theorem
Exponentials and Logarithms Revisited
Differentiation and Limits
Constant multiples of functions
Sums and differences of functions
Proof of the product rule
Proof of the quotient rule
Proof of the chain rule
Proof of the Extreme Value Theorem
Proof of Rolle's Theorem
Proof of the Mean Value Theorem
The error in linearization
Derivatives of piecewise-defined functions
Proof of l'Hopital's Rule
Proof of the Taylor Approximation Theorem
Estimating Integrals
Estimating Integrals Using Strips
Evenly spaced partitions
The Trapezoidal Rule
Simpson's Rule
Proof of Simpson's rule
The Error in Our Approximations
Examples of estimating the error
Proof of an error term inequality
List of Symbols
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