
Prerequisites for Calculus 


2  (56) 


3  (9) 





Parallel and Perpendicular Lines 








12  (10) 





Viewing and Interpreting Graphs 



Even Functions and Odd FunctionsSymmetry 



Functions Defined in Pieces 








22  (8) 










30  (7) 










37  (9) 














46  (12) 



Graphs of Trigonometric Functions 





Even and Odd Trigonometric Functions 



Transformations of Trigonometric Graphs 



Inverse Trigonometric Functions 




55  (1) 


56  (2) 


58  (40) 

Rates of Change and Limits 


59  (11) 

Average and Instantaneous Speed 







Onesided and Twosided Limits 





Limits Involving Infinity 


70  (8) 



Sandwich Theorem Revisited 







``Seeing'' Limits as x → ± ∞ 




78  (9) 









Intermediate Value Theorem for Continuous Functions 



Rates of Change and Tangent Lines 


87  (11) 












95  (1) 


95  (3) 


98  (88) 


99  (10) 

Definition of a Derivative 





Relationship Between the Graphs of f and f' 



Graphing the Derivative from Data 






109  (7) 

How f'(a) Might Fail to Exist 



Differentiability Implies Local Linearity 



Derivatives on a Calculator 



Differentiability Implies Continuity 



Intermediate Value Theorem for Derivatives 



Rules for Differentiation 


116  (11) 

Positive Integer Powers, Multiples, Sums, and Differences 





Negative Integer Powers of x 



Second and Higher Order Derivatives 



Velocity and Other Rates of Change 


127  (14) 

Instantaneous Rates of Change 









Derivatives of Trigonometric Functions 


141  (7) 

Derivative of the Sine Function 



Derivative of the Cosine Function 







Derivatives of Other Basic Trigonometric Functions 




148  (9) 

Derivative of a Composite Function 





Repeated Use of the Chain Rule 



Slopes of Parametrized Curves 






157  (8) 

Implicitly Defined Functions 



Lenses, Tangents, and Normal Lines 



Derivatives of Higher Order 



Rational Powers of Differentiable Functions 



Derivatives of Inverse Trigonometric Functions 


165  (7) 

Derivatives of Inverse Functions 



Derivative of the Arcsine 



Derivative of the Arctangent 



Derivative of the Arcsecant 



Derivatives of the Other Three 



Derivatives of Exponential and Logarithmic Functions 


172  (14) 









Power Rule for Arbitrary Real Powers 




181  (1) 


181  (1) 


181  (5) 

Applications of Derivatives 


186  (76) 

Extreme Values of Functions 


187  (9) 

Absolute (Global) Extreme Values 



Local (Relative) Extreme Values 






196  (9) 





Increasing and Decreasing Functions 





Connecting f' and f'' with the Graph of f 


205  (14) 

First Derivative Test for Local Extrema 







Second Derivative Test for Local Extrema 



Learning about Functions from Derivatives 



Modeling and Optimization 


219  (14) 

Examples from Mathematics 



Examples from Business and Industry 





Modeling Discrete Phenomena with Differentiable Functions 



Linearization and Newton's Method 


233  (13) 







Estimating Change with Differentials 



Absolute, Relative, and Percentage Change 






246  (16) 





Simulating Related Motion 




255  (1) 


256  (6) 


262  (58) 

Estimating with Finite Sums 


263  (11) 



Rectangular Approximation Method (RAM) 








274  (11) 



Terminology and Notation of Integration 



Definite Integral and Area 





Integrals on a Calculator 



Discontinuous Integrable Functions 



Definite Integrals and Antiderivatives 


285  (9) 

Properties of Definite Integrals 



Average Value of a Function 



Mean Value Theorem for Definite Integrals 



Connecting Differential and Integral Calculus 



Fundamental Theorem of Calculus 


294  (12) 

Fundamental Theorem, Part 1 



Graphing the Function ∫ x/α f(t)dt 



Fundamental Theorem, Part 2 





Analyzing Antiderivatives Graphically 




306  (14) 

Trapezoidal Approximations 








315  (1) 


315  (4) 


319  (1) 

Differential Equations and Mathematical Modeling 


320  (58) 

Slope Fields and Euler's Method 


321  (10) 







Antidifferentiation by Substitution 


331  (10) 



Leibniz Notation and Antiderivatives 



Substitution in Indefinite Integrals 



Substitution in Definite Integrals 



Antidifferentiation by Parts 


341  (9) 

Product Rule in Integral Form 



Solving for the Unknown Integral 





Inverse Trigonometric and Logarithmic Functions 



Exponential Growth and Decay 


350  (12) 

Separable Differential Equations 



Law of Exponential Change 



Continuously Compounded Interest 





Modeling Growth with Other Bases 






362  (16) 





The Logistic Differential Equation 






372  (1) 


372  (4) 


376  (2) 

Applications of Definite Integrals 


378  (41) 


379  (11) 












390  (9) 



Area Enclosed by Intersecting Curves 



Boundaries with Changing Functions 



Integrating with Respect to y 



Saving Time with Geometry Formulas 




399  (13) 












412  (7) 





Vertical Tangents, Corners, and Cusps 



Applications from Science and Statistics 


419  (15) 



Fluid Force and Fluid Pressure 






430  (1) 


430  (1) 


430  (4) 

Sequences, L'Hopital's Rule, and Improper Integrals 


434  (38) 


435  (9) 



Arithmetic and Geometric Sequences 








444  (9) 



Indeterminate Forms ∞/∞, ∞ 0, and ∞  ∞ 



Indeterminate Forms 1∞, 00, ∞0 




453  (6) 

Comparing Rates of Growth 



Using L'Hopital's Rule to Compare Growth Rates 



Sequential versus Binary Search 




459  (13) 

Infinite Limits of Integration 



Integrands with Infinite Discontinuities 



Test for Convergence and Divergence 






470  (1) 


470  (2) 


472  (58) 


473  (11) 



Representing Functions by Series 



Differentiation and Integration 






484  (11) 



Series for sin x and cos x 





Maclaurin and Taylor Series 





Table of Maclaurin Series 




495  (8) 





Remainder Estimation Theorem 






503  (10) 





Comparing Nonnegative Series 







Testing Convergence at Endpoints 


513  (17) 



Harmonic Series and pseries 







Absolute and Conditional Convergence 








526  (1) 


526  (3) 


529  (1) 

Parametric, Vector, and Polar Functions 


530  (88) 


531  (7) 

Parametric Curves in the Plane 










538  (10) 







Velocity, Acceleration, and Speed 



Displacement and Distance Traveled 




548  (14) 







Areas Enclosed by Polar Curves 






559  (1) 


560  (2) 



Formulas from Precalculus Mathematics 


562  (4) 


566  (3) 

Using the Limit Definition 


569  (8) 


577  (1) 


578  (25) 


603  (9) 

A Brief Table of Integrals 


612  (6) 
Glossary 

618  (11) 
Selected Answers 

629  (51) 
Applications Index 

680  (4) 
Index 

684  