The everything guide to calculus I : a step-by-step guide to the basics of calculus-- in plain English!

Calculus is the basis of all advanced science and math. But it can be very intimidating, especially if you're learning it for the first time! If finding derivatives or understanding integrals has you stumped, this book can guide you through it. This indispensable resource offers hundreds of practice exercises and covers all the key concepts of calculus, including: Limits of a function Derivatives of a function Monomials and polynomials Calculating maxima and minima Logarithmic differentials Integrals Finding the volume of irregularly shaped objects By breaking down challenging concepts and presenting clear explanations, you'll solidify your knowledge base--and face calculus without fear! Contents......Page 6 Acknowledgments......Page 10 The Top 10 Ways to Be Successful in Calculus......Page 11 Introduction......Page 12 01 Prerequisite Skills......Page 14 Important Algebra Skills......Page 15 The Geometry of Calculus......Page 22 A Bit of Trigonometry......Page 24 Functions and Composition......Page 29 02 Start Building with Limits......Page 32 Foundation of Calculus......Page 33 Concept of a Limit......Page 34 One-Sided Limits......Page 35 Limit at a Point......Page 37 Limits As x Approaches Infinity......Page 44 Skill Check......Page 48 03 The Importance of Continuity......Page 50 Continuity at a Point......Page 51 Kinds of Discontinuity......Page 53 Continuity on an Interval......Page 57 Skill Check......Page 58 04 Getting Differentiability Straight......Page 60 Average Rate of Change......Page 61 Instantaneous Rate of Change......Page 63 Tangent Lines......Page 66 Definition of the Derivative......Page 69 Conditions for Differentiability......Page 71 Skill Check......Page 73 05 Derivatives of Polynomials......Page 74 A Second Definition of the Derivative......Page 75 Standard Notations......Page 76 Derivative at Any Point......Page 77 Derivative Rules for Monomials and Polynomials......Page 80 Derivatives of Products and Quotients......Page 83 Skill Check......Page 86 06 Derivatives of Trigonometric Functions......Page 87 Derivatives on a Graphing Calculator......Page 88 Derivative of the Sine Function......Page 90 Derivative of the Cosine Function......Page 92 Derivative of the Tangent Function......Page 93 Using Trigonometric Identities......Page 96 Skill Check......Page 97 07 The Chain Rule......Page 99 Understanding the Notation......Page 100 Applying the Chain Rule to Powers......Page 103 Implicit Differentiation......Page 106 Skill Check......Page 109 08 Derivatives of Other Functions......Page 111 The Derivative of ln(x)......Page 112 The Derivative of loga x......Page 114 Logarithmic Differentiation......Page 115 The Derivative of ax......Page 116 Skill Check......Page 118 09 Derivatives of Inverse Trigonometric Functions......Page 119 Inverse Functions......Page 120 Derivatives of Inverse Functions......Page 122 Derivative of the Inverse Sine Function......Page 124 Derivatives of the Other Inverse Functions......Page 125 Skill Check......Page 127 10 Higher-Order Derivatives......Page 128 What the Second Derivative Means......Page 129 Implications for Particle Motion......Page 130 Higher Derivatives of Explicit Functions......Page 131 Second Derivatives of Implicit Functions......Page 135 Skill Check......Page 136 11 Graph Analysis Using Derivatives......Page 138 Producing a Graph of a First Derivative......Page 139 Sketching a Function Using Its Derivative......Page 142 Producing a More Detailed Graph......Page 145 Skill Check......Page 149 12 Applications of Derivatives......Page 151 Local Maxima and Minima......Page 152 Absolute Extrema......Page 155 Optimization......Page 157 Inflection Points......Page 159 The Mean Value Theorem......Page 161 Linear Motion......Page 163 Related Rates......Page 165 Skill Check......Page 168 13 Area by Numerical Methods......Page 170 Area under a Graph......Page 171 Riemann Sums......Page 172 The Definition of a Definite Integral......Page 175 The Trapezoidal Rule......Page 178 Simpson’s Rule......Page 180 Integrals on a Graphing Calculator......Page 182 Skill Check......Page 183 14 The Definite Integral Explored......Page 185 Area for Negative Functions......Page 186 Switching Limits......Page 187 Four More Basic Properties......Page 188 Net Area......Page 191 Skill Check......Page 193 15 The Fundamental Theorem of Calculus......Page 195 Integral as a Function......Page 196 Antiderivatives......Page 197 Rate of Change of an Integral......Page 200 Evaluation of Integrals......Page 204 Skill Check......Page 205 16 Methods of Antidifferentiation......Page 207 Geometric Methods......Page 208 Changing the Integrand Using Algebra......Page 210 Substitution......Page 212 Integration by Parts......Page 215 Additional Methods......Page 217 Skill Check......Page 218 17 Indefinite Integrals......Page 219 General and Specific Solutions......Page 220 Slopefields......Page 222 Exponential Growth......Page 225 Logistic Growth......Page 227 Skill Check......Page 229 18 The Integral as an Accumulator......Page 230 Accumulation......Page 231 Net Change in a Quantity......Page 232 Average Value......Page 233 Total Distance and Displacement......Page 236 Skill Check......Page 238 19 Applications of Integrals......Page 240 Area Between Curves......Page 241 Finding a Volume by Cross Sections......Page 244 Finding a Volume by Discs......Page 247 Finding a Volume by Washers......Page 248 Arc Length......Page 251 Skill Check......Page 254 Appendix A: Useful Prerequisite Information......Page 256 Appendix B: Derivatives and Integrals......Page 264 Appendix C: The Final Exam......Page 268 Appendix D: Answer Key......Page 275 Index......Page 313 Contents 6 Acknowledgments 10 The Top 10 Ways to Be Successful in Calculus 11 Introduction 12 01 Prerequisite Skills 14 Important Algebra Skills 15 The Geometry of Calculus 22 A Bit of Trigonometry 24 Nine Basic Functions 29 Functions and Composition 29 02 Start Building with Limits 32 Foundation of Calculus 33 Concept of a Limit 34 One-Sided Limits 35 Limit at a Point 37 Limits As x Approaches Infinity 44 Skill Check 48 03 The Importance of Continuity 50 Continuity at a Point 51 Kinds of Discontinuity 53 Continuity on an Interval 57 The Intermediate Value Theorem 58 Skill Check 58 04 Getting Differentiability Straight 60 Average Rate of Change 61 Instantaneous Rate of Change 63 Tangent Lines 66 Definition of the Derivative 69 Conditions for Differentiability 71 Skill Check 73 05 Derivatives of Polynomials 74 A Second Definition of the Derivative 75 Standard Notations 76 Derivative at Any Point 77 Derivative Rules for Monomials and Polynomials 80 Derivatives of Products and Quotients 83 Skill Check 86 06 Derivatives of Trigonometric Functions 87 Derivatives on a Graphing Calculator 88 Derivative of the Sine Function 90 Derivative of the Cosine Function 92 Derivative of the Tangent Function 93 Using Trigonometric Identities 96 Skill Check 97 07 The Chain Rule 99 Derivatives of Composite Functions 100 Understanding the Notation 100 Applying the Chain Rule to Powers 103 Implicit Differentiation 106 Skill Check 109 08 Derivatives of Other Functions 111 The Derivative of ex 112 The Derivative of ln(x) 112 The Derivative of loga x 114 Logarithmic Differentiation 115 The Derivative of ax 116 Skill Check 118 09 Derivatives of Inverse Trigonometric Functions 119 Inverse Functions 120 Derivatives of Inverse Functions 122 Derivative of the Inverse Sine Function 124 Derivatives of the Other Inverse Functions 125 Skill Check 127 10 Higher-Order Derivatives 128 Notation 129 What the Second Derivative Means 129 Implications for Particle Motion 130 Higher Derivatives of Explicit Functions 131 Second Derivatives of Implicit Functions 135 Skill Check 136 11 Graph Analysis Using Derivatives 138 Curve Sketching 139 Producing a Graph of a First Derivative 139 Sketching a Function Using Its Derivative 142 Producing a More Detailed Graph 145 Skill Check 149 12 Applications of Derivatives 151 Local Maxima and Minima 152 Absolute Extrema 155 Optimization 157 Inflection Points 159 The Mean Value Theorem 161 Linear Motion 163 Related Rates 165 Skill Check 168 13 Area by Numerical Methods 170 Area under a Graph 171 Riemann Sums 172 The Definition of a Definite Integral 175 The Trapezoidal Rule 178 Simpson’s Rule 180 Integrals on a Graphing Calculator 182 Skill Check 183 14 The Definite Integral Explored 185 Area for Negative Functions 186 Switching Limits 187 Four More Basic Properties 188 Net Area 191 Skill Check 193 15 The Fundamental Theorem of Calculus 195 Integral as a Function 196 Antiderivatives 197 Rate of Change of an Integral 200 Evaluation of Integrals 204 Skill Check 205 16 Methods of Antidifferentiation 207 Geometric Methods 208 Changing the Integrand Using Algebra 210 Substitution 212 Integration by Parts 215 Additional Methods 217 Skill Check 218 17 Indefinite Integrals 219 Differential Equations 220 General and Specific Solutions 220 Slopefields 222 Exponential Growth 225 Logistic Growth 227 Skill Check 229 18 The Integral as an Accumulator 230 Accumulation 231 Net Change in a Quantity 232 Average Value 233 Total Distance and Displacement 236 Skill Check 238 19 Applications of Integrals 240 Area Between Curves 241 Finding a Volume by Cross Sections 244 Finding a Volume by Discs 247 Finding a Volume by Washers 248 Arc Length 251 Skill Check 254 Appendix A: Useful Prerequisite Information 256 Appendix B: Derivatives and Integrals 264 Appendix C: The Final Exam 268 Appendix D: Answer Key 275 Index 313 The top 10 ways to be successful in calculus Prerequisite skills Start building with limits The importance of continuity Getting differentiability straight Derivatives of polynomials Derivatives of trigonometric functions The chain rule Derivatives of other functions Derivatives of inverse trigonometric functions Higher-order derivatives Graph analysis using derivatives Applications of derivatives Area by numerical methods The definitive integral explored The fundamental theorem of calculus Methods of antidifferentiation Indefinite integrals The integral as an accumulator Applications of integrals.