The Calculus Lifesaver: All the Tools You Need to Excel at Calculus (Princeton Lifesaver Study Guides)

For many students, calculus can be the most mystifying and frustrating course they will ever take. The Calculus Lifesaver provides 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 Lifesaver combines 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 — is available at Adrian Banner lectures - 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 Cover Contents 1 Functions, Graphs, and Lines 1.1 Functions 1.1.1 Interval notation 1.1.2 Finding the domain 1.1.3 Finding the range using the graph 1.1.4 The vertical line test 1.2 Inverse Functions 1.2.1 The horizontal line test 1.2.2 Finding the inverse 1.2.3 Restricting the domain 1.2.4 Inverses of inverse functions 1.3 Composition of Functions 1.4 Odd and Even Functions 1.5 Graphs of Linear Functions 1.6 Common Functions and Graphs 2 Review of Trigonometry 2.1 The Basics 2.2 Extending the Domain of Trig Functions 2.2.1 The ASTC method 2.2.2 Trig functions outside [0, 2π] 2.3 The Graphs of Trig Functions 2.4 Trig Identities 3 Introduction to Limits 3.1 Limits: The Basic Idea 3.2 Left-Hand and Right-Hand Limits 3.3 When the Limit Does Not Exist 3.4 Limits at ∞ and −∞ 3.4.1 Large numbers and small numbers 3.5 Two Common Misconceptions about Asymptotes 3.6 The Sandwich Principle 3.7 Summary of Basic Types of Limits 4 How to Solve Limit Problems Involving Polynomials 4.1 Limits Involving Rational Functions as x → a 4.2 Limits Involving Square Roots as x → a 4.3 Limits Involving Rational Functions as x → ∞ 4.3.1 Method and examples 4.4 Limits Involving Poly-type Functions as x → ∞ 4.5 Limits Involving Rational Functions as x → −∞ 4.6 Limits Involving Absolute Values 5 Continuity and Differentiability 5.1 Continuity 5.1.1 Continuity at a point 5.1.2 Continuity on an interval 5.1.3 Examples of continuous functions 5.1.4 The Intermediate Value Theorem 5.1.5 A harder IVT example 5.1.6 Maxima and minima of continuous functions 5.2 Differentiability 5.2.1 Average speed 5.2.2 Displacement and velocity 5.2.3 Instantaneous velocity 5.2.4 The graphical interpretation of velocity 5.2.5 Tangent lines 5.2.6 The derivative function 5.2.7 The derivative as a limiting ratio 5.2.8 The derivative of linear functions 5.2.9 Second and higher-order derivatives 5.2.10 When the derivative does not exist 5.2.11 Differentiability and continuity 6 How to Solve Differentiation Problems 6.1 Finding Derivatives Using the De nition 6.2 Finding Derivatives (the Nice Way) 6.2.1 Constant multiples of functions 6.2.2 Sums and differences of functions 6.2.3 Products of functions via the product rule 6.2.4 Quotients of functions via the quotient rule 6.2.5 Composition of functions via the chain rule 6.2.6 A nasty example 6.2.7 Justification of the product rule and the chain rule 6.3 Finding the Equation of a Tangent Line 6.4 Velocity and Acceleration 6.4.1 Constant negative acceleration 6.5 Limits Which Are Derivatives in Disguise 6.6 Derivatives of Piecewise-Defined Functions 6.7 Sketching Derivative Graphs Directly 7 Trig Limits and Derivatives 7.1 Limits Involving Trig Functions 7.1.1 The small case 7.1.2 Solving problems—the small case 7.1.3 The large case 7.1.4 The "other" case 7.1.5 Proof of an important limit 7.2 Derivatives Involving Trig Functions 7.2.1 Examples of differentiating trig functions 7.2.2 Simple harmonic motion 7.2.3 A curious function 8 Implicit Differentiation and Related Rates 8.1 Implicit Differentiation 8.1.1 Techniques and examples 8.1.2 Finding the second derivative implicitly 8.2 Related Rates 8.2.1 A simple example 8.2.2 A slightly harder example 8.2.3 A much harder example 8.2.4 A really hard example 9 Exponentials and Logarithms 9.1 The Basics 9.1.1 Review of exponentials 9.1.2 Review of logarithms 9.1.3 Logarithms, exponentials, and inverses 9.1.4 Log rules 9.2 Definition of e 9.2.1 A question about compound interest 9.2.2 The answer to our question 9.2.3 More about e and logs 9.3 Differentiation of Logs and Exponentials 9.3.1 Examples of differentiating exponentials and logs 9.4 How to Solve Limit Problems Involving Exponentials or Logs 9.4.1 Limits involving the definition of e 9.4.2 Behavior of exponentials near 0 9.4.3 Behavior of logarithms near 1 9.4.4 Behavior of exponentials near ∞ or −∞ 9.4.5 Behavior of logs near ∞ 9.4.6 Behavior of logs near 0 9.5 Logarithmic Differentiation 9.5.1 The derivative of x^a 9.6 Exponential Growth and Decay 9.6.1 Exponential growth 9.6.2 Exponential decay 9.7 Hyperbolic Functions 10 Inverse Functions and Inverse Trig Functions 10.1 The Derivative and Inverse Functions 10.1.1 Using the derivative to show that an inverse exists 10.1.2 Derivatives and inverse functions: what can go wrong 10.1.3 Finding the derivative of an inverse function 10.1.4 A big example 10.2 Inverse Trig Functions 10.2.1 Inverse sine 10.2.2 Inverse cosine 10.2.3 Inverse tangent 10.2.4 Inverse secant 10.2.5 Inverse cosecant and inverse cotangent 10.2.6 Computing inverse trig functions 10.3 Inverse Hyperbolic Functions 10.3.1 The rest of the inverse hyperbolic functions 11 The Derivative and Graphs 11.1 Extrema of Functions 11.1.1 Global and local extrema 11.1.2 The Extreme Value Theorem 11.1.3 How to find global maxima and minima 11.2 Rolle's Theorem 11.3 The Mean Value Theorem 11.3.1 Consequences of the Mean Value Theorem 11.4 The Second Derivative and Graphs 11.4.1 More about points of inflection 11.5 Classifying Points Where the Derivative Vanishes 11.5.1 Using the first derivative 11.5.2 Using the second derivative 12 Sketching Graphs 12.1 How to Construct a Table of Signs 12.1.1 Making a table of signs for the derivative 12.1.2 Making a table of signs for the second derivative 12.2 The Big Method 12.3 Examples 12.3.1 An example without using derivatives 12.3.2 The full method: example 1 12.3.3 The full method: example 2 12.3.4 The full method: example 3 12.3.5 The full method: example 4 13 Optimization and Linearization 13.1 Optimization 13.1.1 An easy optimization example 13.1.2 Optimization problems: the general method 13.1.3 An optimization example 13.1.4 Another optimization example 13.1.5 Using implicit differentiation in optimization 13.1.6 A difficult optimization example 13.2 Linearization 13.2.1 Linearization in general 13.2.2 The differential 13.2.3 Linearization summary and examples 13.2.4 The error in our approximation 13.3 Newton's Method 14 L'Hôpital's Rule and Overview of Limits 14.1 L'Hôpital's Rule 14.1.1 Type A: 0/0 case 14.1.2 Type A: ±∞/±∞ case 14.1.3 Type B1 (∞ − ∞) 14.1.4 Type B2 (0 × ±∞) 14.1.5 Type C (1^±∞, 0^0, or ∞^0) 14.1.6 Summary of l'Hôpital's Rule types 14.2 Overview of Limits 15 Introduction to Integration 15.1 Sigma Notation 15.1.1 A nice sum 15.1.2 Telescoping series 15.2 Displacement and Area 15.2.1 Three simple cases 15.2.2 A more general journey 15.2.3 Signed area 15.2.4 Continuous velocity 15.2.5 Two special approximations 16 Definite Integrals 16.1 The Basic Idea 16.1.1 Some easy examples 16.2 Definition of the Definite Integral 16.2.1 An example of using the definition 16.3 Properties of Definite Integrals 16.4 Finding Areas 16.4.1 Finding the unsigned area 16.4.2 Finding the area between two curves 16.4.3 Finding the area between a curve and the y-axis 16.5 Estimating Integrals 16.5.1 A simple type of estimation 16.6 Averages and the Mean Value Theorem for Integrals 16.6.1 The Mean Value Theorem for integrals 16.7 A Nonintegrable Function 17 The Fundamental Theorems of Calculus 17.1 Functions Based on Integrals of Other Functions 17.2 The First Fundamental Theorem 17.2.1 Introduction to antiderivatives 17.3 The Second Fundamental Theorem 17.4 Indefinite Integrals 17.5 How to Solve Problems: The First Fundamental Theorem 17.5.1 Variation 1: variable left-hand limit of integration 17.5.2 Variation 2: one tricky limit of integration 17.5.3 Variation 3: two tricky limits of integration 17.5.4 Variation 4: limit is a derivative in disguise 17.6 How to Solve Problems: The Second Fundamental Theorem 17.6.1 Finding indefinite integrals 17.6.2 Finding definite integrals 17.6.3 Unsigned areas and absolute values 17.7 A Technical Point 17.8 Proof of the First Fundamental Theorem 18 Techniques of Integration, Part One 18.1 Substitution 18.1.1 Substitution and definite integrals 18.1.2 How to decide what to substitute 18.1.3 Theoretical justification of the substitution method 18.2 Integration by Parts 18.2.1 Some variations 18.3 Partial Fractions 18.3.1 The algebra of partial fractions 18.3.2 Integrating the pieces 18.3.3 The method and a big example 19 Techniques of Integration, Part Two 19.1 Integrals Involving Trig Identities 19.2 Integrals Involving Powers of Trig Functions 19.2.1 Powers of sin and/or cos 19.2.2 Powers of tan 19.2.3 Powers of sec 19.2.4 Powers of cot 19.2.5 Powers of csc 19.2.6 Reduction formulas 19.3 Integrals Involving Trig Substitutions 19.3.1 Type 1: sqrt(a^2 - x^2) 19.3.2 Type 2: sqrt(x^2 + a^2) 19.3.3 Type 3: sqrt(x^2 - a^2) 19.3.4 Completing the square and trig substitutions 19.3.5 Summary of trig substitutions 19.3.6 Technicalities of square roots and trig substitutions 19.4 Overview of Techniques of Integration 20 Improper Integrals: Basic Concepts 20.1 Convergence and Divergence 20.1.1 Some examples of improper integrals 20.1.2 Other blow-up points 20.2 Integrals over Unbounded Regions 20.3 The Comparison Test (Theory) 20.4 The Limit Comparison Test (Theory) 20.4.1 Functions asymptotic to each other 20.4.2 The statement of the test 20.5 The p-test (Theory) 20.6 The Absolute Convergence Test 21 Improper Integrals: How to Solve Problems 21.1 How to Get Started 21.1.1 Splitting up the integral 21.1.2 How to deal with negative function values 21.2 Summary of Integral Tests 21.3 Behavior of Common Functions near ∞ and −∞ 21.3.1 Polynomials and poly-type functions near ∞ and −∞ 21.3.2 Trig functions near ∞ and −∞ 21.3.3 Exponentials near ∞ and −∞ 21.3.4 Logarithms near ∞ 21.4 Behavior of Common Functions near 0 21.4.1 Polynomials and poly-type functions near 0 21.4.2 Trig functions near 0 21.4.3 Exponentials near 0 21.4.4 Logarithms near 0 21.4.5 The behavior of more general functions near 0 21.5 How to Deal with Problem Spots Not at 0 or ∞ 22 Sequences and Series: Basic Concepts 22.1 Convergence and Divergence of Sequences 22.1.1 The connection between sequences and functions 22.1.2 Two important sequences 22.2 Convergence and Divergence of Series 22.2.1 Geometric series (theory) 22.3 The nth Term Test (Theory) 22.4 Properties of Both In nite Series and Improper Integrals 22.4.1 The comparison test (theory) 22.4.2 The limit comparison test (theory) 22.4.3 The p-test (theory) 22.4.4 The absolute convergence test 22.5 New Tests for Series 22.5.1 The ratio test (theory) 22.5.2 The root test (theory) 22.5.3 The integral test (theory) 22.5.4 The alternating series test (theory) 23 How to Solve Series Problems 23.1 How to Evaluate Geometric Series 23.2 How to Use the nth Term Test 23.3 How to Use the Ratio Test 23.4 How to Use the Root Test 23.5 How to Use the Integral Test 23.6 Comparison Test, Limit Comparison Test, and p-test 23.7 How to Deal with Series with Negative Terms 24 Taylor Polynomials, Taylor Series, and Power Series 24.1 Approximations and Taylor Polynomials 24.1.1 Linearization revisited 24.1.2 Quadratic approximations 24.1.3 Higher-degree approximations 24.1.4 Taylor's Theorem 24.2 Power Series and Taylor Series 24.2.1 Power series in general 24.2.2 Taylor series and Maclaurin series 24.2.3 Convergence of Taylor series 24.3 A Useful Limit 25 How to Solve Estimation Problems 25.1 Summary of Taylor Polynomials and Series 25.2 Finding Taylor Polynomials and Series 25.3 Estimation Problems Using the Error Term 25.3.1 First example 25.3.2 Second example 25.3.3 Third example 25.3.4 Fourth example 25.3.5 Fifth example 25.3.6 General techniques for estimating the error term 25.4 Another Technique for Estimating the Error 26 Taylor and Power Series: How to Solve Problems 26.1 Convergence of Power Series 26.1.1 Radius of convergence 26.1.2 How to find the radius and region of convergence 26.2 Getting New Taylor Series from Old Ones 26.2.1 Substitution and Taylor series 26.2.2 Differentiating Taylor series 26.2.3 Integrating Taylor series 26.2.4 Adding and subtracting Taylor series 26.2.5 Multiplying Taylor series 26.2.6 Dividing Taylor series 26.3 Using Power and Taylor Series to Find Derivatives 26.4 Using Maclaurin Series to Find Limits 27 Parametric Equations and Polar Coordinates 27.1 Parametric Equations 27.1.1 Derivatives of parametric equations 27.2 Polar Coordinates 27.2.1 Converting to and from polar coordinates 27.2.2 Sketching curves in polar coordinates 27.2.3 Finding tangents to polar curves 27.2.4 Finding areas enclosed by polar curves 28 Complex Numbers 28.1 The Basics 28.1.1 Complex exponentials 28.2 The Complex Plane 28.2.1 Converting to and from polar form 28.3 Taking Large Powers of Complex Numbers 28.4 Solving z^n = w 28.4.1 Some variations 28.5 Solving e^z = w 28.6 Some Trigonometric Series 28.7 Euler's Identity and Power Series 29 Volumes, Arc Lengths, and Surface Areas 29.1 Volumes of Solids of Revolution 29.1.1 The disc method 29.1.2 The shell method 29.1.3 Summary 29.1.4 Variation 1: regions between a curve and the y-axis 29.1.5 Variation 2: regions between two curves 29.1.6 Variation 3: axes parallel to the coordinate axes 29.2 Volumes of General Solids 29.3 Arc Lengths 29.3.1 Parametrization and speed 29.4 Surface Areas of Solids of Revolution 30 Differential Equations 30.1 Introduction to Differential Equations 30.2 Separable First-order Differential Equations 30.3 First-order Linear Equations 30.3.1 Why the integrating factor works 30.4 Constant-coefficient Differential Equations 30.4.1 Solving first-order homogeneous equations 30.4.2 Solving second-order homogeneous equations 30.4.3 Why the characteristic quadratic method works 30.4.4 Nonhomogeneous equations and particular solutions 30.4.5 Finding a particular solution 30.4.6 Examples of finding particular solutions 30.4.7 Resolving conflicts between yP and yH 30.4.8 Initial value problems (constant-coefficient linear) 30.5 Modeling Using Differential Equations Appendix A Limits and Proofs A.1 Formal Definition of a Limit A.1.1 A little game A.1.2 The actual definition A.1.3 Examples of using the definition A.2 Making New Limits from Old Ones A.2.1 Sums and differences of limits—proofs A.2.2 Products of limits—proof A.2.3 Quotients of limits—proof A.2.4 The sandwich principle—proof A.3 Other Varieties of Limits A.3.1 Infinite limits A.3.2 Left-hand and right-hand limits A.3.3 Limits at ∞ and −∞ A.3.4 Two examples involving trig A.4 Continuity and Limits A.4.1 Composition of continuous functions A.4.2 Proof of the Intermediate Value Theorem A.4.3 Proof of the Max-Min Theorem A.5 Exponentials and Logarithms Revisited A.6 Differentiation and Limits A.6.1 Constant multiples of functions A.6.2 Sums and differences of functions A.6.3 Proof of the product rule A.6.4 Proof of the quotient rule A.6.5 Proof of the chain rule A.6.6 Proof of the Extreme Value Theorem A.6.7 Proof of Rolle's Theorem A.6.8 Proof of the Mean Value Theorem A.6.9 The error in linearization A.6.10 Derivatives of piecewise-defined functions A.6.11 Proof of l'Hôpital's Rule A.7 Proof of the Taylor Approximation Theorem Appendix B Estimating Integrals B.1 Estimating Integrals Using Strips B.1.1 Evenly spaced partitions B.2 The Trapezoidal Rule B.3 Simpson's Rule B.3.1 Proof of Simpson's rule B.4 The Error in Our Approximations B.4.1 Examples of estimating the error B.4.2 Proof of an error term inequality List of Symbols Index For many students, calculus can be the most mystifying and frustrating course they will ever take. Based upon Adrian Banner's popular calculus review course at Princeton University, this book provides students with the essential tools they need not only to learn calculus, but also to excel at it "Finally, a calculus book you can pick up and actually read! Developed especially for students who are motivated to earn an A but only score average grades on exams, The Calculus Lifesaver has all the essentials you need to master calculus."--Jacket