; CALCULUS Al Shenk, Jorge Calvo, Jim Coykendall (The page numbers are for sample Chapters 1, 2, 6, and 14.)

Introduction ...... p. xi To the instructor ...... p. xiii

Chapter 0. Mathematical Models: Functions and Graphs 0.1 Measuring and modeling: variables and functions Variables, functions, and graphs. Functions given by formulas, tables, and graphs. Finding formulas for mathematical models. When is a curve the graph of a function? Change, percent change, and relative change. 0.2 Set notation and solving inequalities Intervals and other sets of numbers. The absolute value function. Working with inequalities. 0.3 Power and exponential functions n Power functions y = x and their graphs. Vertical and horizontal translation. Reflection, x magnification, and contraction. Exponential functions y = b and their graphs. Laws of exponents. Historical notes. 0.4 Inverse functions and logarithms Inverse functions. Changing variables. Restricting domains. Logarithms. The common and natural logarithms. Laws of logarithms. 0.5 Trigonometric and inverse trigonometric functions Radian measure. Trigonometric functions and the unit circle. A direct definition of the tangent function. The graphs of the secant, cosecant, tangent, and cotangent functions. The inverse sine, cosine, tangent, and cotangent functions and their graphs. The Law of Cosines. The Law of Sines. More trigonometric identities. 0.6 Linear combinations, products, quotients, and compositions Linear combinations, products, and quotients of functions. Polynomial and rational functions. Composite functions. 0.R Review Exercises

Chapter 1. Limits and Continuity 1.1 Finite limits ...... p. 1 A first look at instantaneous velocity. One-sided and two-sided finite limits. A function without a one-sided limit. Limits of sums, products, powers, and quotients. Limits of polynomials and rational functions. 1.2 More on finite limits ...... p. 14 Finding limits of quotients when the numerators and denominators tend to zero. Ratio- nalizing differences of square roots 1.3 Continuity ...... p. 20 Continuity at a point. Continuity of polynomials and rational functions. Continuity on intervals. Continuity of functions given by single formulas. The Intermediate Value Theorem. 1.4 Limits involving infinity ...... p. 36 Infinite limits as x . Finite limits as x . One-sided and two-sided infinite → ±∞ → ±∞ limits. Infinite limits of transcendental functions.i p.ii (3/24/07) Table of Contents

1.5 Formal definitions of limits ...... p. 51 The δ-definition of a finite two-sided limit. Using approximate inequalities. Two short proofs. Using graphs with finite limits.. 1.R Review Exercises ...... p. 62

Chapter 2. The Derivative and Applications 2.1 Linear functions and constant rates of change ...... p. 65 Constant velocity. Other constant rates of change. Approximating data with linear functions. 2.2 Average rates of change ...... p. 78 Average velocity. Other average rates of change. Estimating instantaneous velocity with average velocity. 2.3 Tangent lines, rates of change, and derivatives ...... p. 89 Tangent lines, derivatives, and instantaneous rates of change. Predicting a derivative by calculating difference quotients. Finding exact derivatives. Equations of tangent lines. The ∆x∆f-formulation of the definition of the derivative. Finding a rate of change from a tangent line. A secant line program. 2.4 Derivatives of power functions and linear combinations ...... p. 107 n Leibniz notation and the differentiation operator. The derivative of y = x . Derivatives of linear combinations of functions. 2.5 Derivatives as functions and estimating derivatives ...... p. 118 The derivative as a function. Estimating derivatives from graphs. Using a graph to estimate where a derivative has a particular value. Estimating a derivative from a table. The term “marginal” in economics. Differentiability and continuity. Higher-order derivatives. Acceleration. 2.6 Derivatives of products and quotients ...... p. 137 n The Product Rule. Related-rate problems. The Quotient Rule. The derivative of y = x for integers n 2 and Mathematical Induction. ≥ 2.7 Derivatives of powers of functions ...... p. 149 The Chain Rule for powers of functions. On the order of operations. 2.8 Linear approximations and differentials ...... p. 157 Tangent line approximations. Differentials. Error estimates. A closer look at linear approximations. History of the derivative. 2.R Review exercises ...... p. 166 Table of Contents (3/24/07) p.iii

Chapter 3. Derivatives of Transcendental Functions† 3.1 The general Chain Rule An example with linear functions. The general Chain Rule. The Chain Rule in narrative problems. Using the Chain Rule with graphs. 3.2 Derivatives of logarithms The derivative of y = logb x and the number e. Derivative of the natural logarithms. Derivatives of natural logarithms of functions. 3.3 Derivatives of exponential functions Growth factors. Modeling with exponential functions. Radioactive decay and half-life. x Compound interest. Interest compounded continuously. The derivatives of y = e , u x u x n y = e ( ), and y = b ( ). The derivative of y = x for irrational n. 3.4 The differential equation dy/dt = ry Relative rates of change. The differential equation dy/dt = ry with constant r. Newton’s Law of heating and cooling. 3.5 Derivatives of the sine and cosine functions Derivatives of y = sin x and y = cos x. The Chain Rule for sines and cosines of functions. Sinusoidal functions. 3.6 Derivatives of the other trigonometric functions Derivatives of the tangent, cotangent, secant, and cosecant functions. The Chain Rule for the tangent, cotangent, secant, and cosecant functions. Derivatives of the inverse sine and cosine functions. The Chain Rule for the inverse sine and cosine functions. Derivatives of the inverse tangent and cotangent functions. The Chain Rule for the inverse tangent and cotangent functions. 3.7 The hyperbolic functions and their derivatives The hyperbolic sine and cosine functions and their graphs. Hyperbolic functions and hyperbolas. The other hyperbolic functions. Inverse hyperbolic functions. 3.R Review exercises

Chapter 4. Graphing and Optimization 4.1 First-Derivative Tests Local maxima and minima: a necessary condition. The Mean Value Theorem. The First-Derivative Test for increasing and decreasing functions. Local maxima and minima: a sufficient condition. 4.2 Sketching graphs of rational functions Sketching graphs of polynomials and other rational functions using First-Derivative Tests. 4.3 Second-Derivative Tests The Second-Derivative Test for concavity. Inflection points. The Second-Derivative Test for local maxima and minima. Using Second-Derivative Tests to analyze graphs. 4.4 Sketching graphs of nonrational functions Functions constructed from powers with restricted domains. Sketching graphs of nonrational functions.

† Formulas for derivatives of logarithms are derived in Section 3.2 using (i) algebraic properties of y = logb x, (ii) the continuity t of y = logb x and (iii) the existence of the limit e = lim (1+1/t) . Facts (ii) and (iii) are established in Section 6.4, where the t→∞ definition of y = ln x as an integral is discussed. p.iv (3/24/07) Table of Contents

4.5 Maxima and minima on intervals The Extreme Value Theorem. Maxima and minima on finite closed intervals. Maxima and minima on other intervals. 4.6 Applied maximum/minimum problems Finding extreme values with direct use of geometric formulas. Using formulas obtained from rates of change. Problems with constraints. 4.R Review exercises

Chapter 5. Further Applications of Derivatives 5.1 L’Hopital’s Rule Indeterminate forms of types 0/0. Predicting limits with tangent line approximations. The Generalized Mean Value Theorem. L’Hopital’s Rule for limits of indeterminate forms of types 0/0, / , 0 , 10, 1∞, 0, and . ∞ ∞ · ∞ ∞ ∞ − ∞ 5.2 More related rate problems Related rate problems using formulas derived from similar and equilateral triangles, the Pythagorean Theorem and other techniques. 5.3 Implicit differentiation Implicitly defined functions. Derivatives of implicitly defined functions. Tangent lines to curves given by equations in x and y. Derivatives of inverse functions. 5.4 Newton’s method Tangent line approximations and Newton’s method. How Newton’s method can fail. 5.R Review exercises

Chapter 6. The Integral and Applications 6.1 Step function rates of change ...... p. 171 Step function rates of change. Approximating continuous rates of change with step functions. 6.2 The definite integral ...... p. 180 The definite integral. Piecewise continuous functions. Integrals and areas. Special Riemann sums. Properties of definite integrals. Unbounded functions. Riemann sum programs. 6.3 The Fundamental Theorem, Part I ...... p. 198 Another look at Section 6.1. The Fundamental Theorem, Part I 6.4 The Fundamental Theorem, Part II ...... p. 209 The Fundamental Theorem, Part II. Another proof of Part I of the Fundamental Theorem. Derivatives of integrals with functions as limits of integration. Defining y = ln x as an integral. 6.5 Integrals of powers and indefinite integrals ...... p. 218 n Integrals of y = x with n = 1. Indefinite integrals. 6 − 6.6 Estimating definite integrals ...... p. 225 Finding approximate integrals from graphs and tables. The Trapezoid Rule. Upper and lower sums. Simpson’s Rule. Error estimates. Table of Contents (3/24/07) p.v

6.7 Integrals involving transcendental functions ...... p. 243 Integrals of y = x−1. Integrals of exponential functions. Integrals of the hyperbolic sine and cosine functions . Integrals involving trigonometric functions. Integrals of y = 1/√a2 x2, − and y = 1/(a2 + x2). 6.8 Integration by substitution ...... p. 254 Integration by substitution. Substitution in definite integrals. Special linear substitutions. Integrals of tangents, cotangents, secants, and cosecants. History of the integral. 6.R Review exercises ...... p. 267

Chapter 7. Additional Applications of Integrals 7.1 More on areas Areas of regions between graphs y = f(x) and y = g(x) and between curves x = f(y) and x = g(y). 7.2 Volumes by disks and washers Volumes of solids of revolution by the methods of disks and washers. Using differentials. 7.3 Volumes by slicing Volumes by the method of slicing. Using differentials. 7.4 Volumes by cylindrical shells Volumes of solids of revolution by the method of cylindrical shells with x- and y-integration. Using differentials. 7.5 Lengths of graphs and areas of surfaces of revolution Lengths of graphs and areas of surfaces of revolution. Using differentials. 7.6 Integrating acceleration and Newton’s law Finding velocity and position from acceleration. Weight and mass. Newton’s law of motion. Constant acceleration. 7.7. Average values of functions Average values of functions. The Mean Value Theorem for integrals. 7.8 One-dimensional density One-dimensional density. Weights and centers of gravity of rods. 7.9 Work, energy, and forces of fuids Work done by a constant force. Work done by a variable force. Finding integrals from Riemann sums. Kinetic energy of point masses. Potential energy. Forces of fluids on flat surfaces. 7.R Review exercises

Chapter 8. Additional Techniques of Integration 8.1. Integration by parts n ax n n Integration by parts. Integrals of y = x e ,y = x cos(ax), and y = x sin(ax). Integrals ax involving logarithms, inverse sines, and inverse tangents. Integrals of y = e sin(bx) and ax y = e cos(bx) 8.2. Integration with trigonometric identities n m n m Integrals of y = sin x cos x with m or n odd and 0. Integrals of y = sin x cos x with ≥ m or n even and 0. Integrals of products of y = sin(ax) and y = cos(bx). Reduction ≥ formulas. p.vi (3/24/07) Table of Contents

8.3. Inverse trigonometric substitutions Inverse sine substitutions. Inverse tangent substitutions. Inverse secant substitutions. Completing squares. A table of integrals. 8.4 Integration by partial fractions Partial fraction decomposition of rational functions. Factoring polynomials. Proper rational functions. Integration by partial fractions with nonrepeating linear factors. Integration by partial fractions—the general case. Integrating improper quotients of polynomials. 8.5 Using integral tables and computer/calculator algebra systems The direct use of integral tables. Using tables with integration by substitution. Using computer or calculator algebra systems. 8.6 Improper integrals Improper integrals with infinite limits of integration. Improper integrals at vertical asymptotes. Improper integrals involving more than one limit. Normal probability distributions. 8.R Review exercises

Chapter 9. Differential Equations 9.1 Separable first-order equations Slope (direction) fields. The differential equation dy/dx = ry with constant r. The differential equation dy/dx = r(x)y. General separable differential equations. Applications to growth and decay. The Logistic Equation of population growth. 9.2 More applications of separable equations Velocity with air resistance. Loan payments. Curves as solutions of first-order differential equations. Predator-prey differential equations. Stimulus and sensation. Orthogonal trajectories. Chemical-reaction rates. 9.3 Linear first-order equations Linear first-order differential equations. Mixing problems. 9.4 Approximate solutions of first-order equations Euler’s Method. A trapezoid method with Euler’s Method as predictor. A Runge-Kutta Method. Euler/Runge-Kutta calculator programs. Slope-field calculator programs.† 9.5 The differential equation ay00 + by0 + cy = 0 Vibrating springs. General solutions of ay00 + by0 + cy = 0. Free, undamped motion of a spring. Free, overdamped motion of a spring. Free, critically damped motion of a spring. Simple electric circuits. 9.6 The differential equation ay00 + by0 + cy = f The Method of Undetermined Coefficients. Variation of parameters. Motion of forced springs. 9.R Review exercises

†These programs for a variety of calculators will be available on the web page for the text. Table of Contents (3/24/07) p.vii

Chapter 10. Sequences and Series 10.1 Infinite sequences Convergence and divergence of infinite sequences. Bounded and monotone sequences. Least upper and greatest lower bounds.‡ 10.2 Geometric series Finite geometric series. Infinite geometric series. Infinite repeating decimals. 10.3 The Integral Test Convergence of series with nonnegative terms. The Integral Test for series with positive terms. The p-series. Absolute convergence. The general Integral Test. Estimating errors with the Integral Test. 10.4 Comparison Tests A divergence test. The Comparison Test. The Limit Comparison Test. 10.5 The Ratio Test and alternating series The Ratio Test. The Ratio Test and p-series. The Alternating-Series Test. Conditional convergence. Estimating errors made with partial sums of alternating series. 10.6 Taylor polynomials and Taylor’s Theorem Taylor polynomials. Taylor’s theorem. Estmating errors. 10.7 Power series Power series. Interval and radius of convergence. Taylor and MacLaurin series. Proving that functions equal their Taylor series. Operations with power series. 10.R Review exercises

Chapter 11. Conic Sections and Polar Coordinates 11.1 Parabolas, ellipses, and hyperbolas Cross sections of cones. Parabolas. Translation of exes. Parabolic trajectories. An optical proerty of parabolas. Ellipses. Hyperbolas. Optical properties of ellipses and hyperbolas. Hyperbolas and navagation. 11.2 Rotation of axes and discriminants Rotating coordinate axes. Classifying conic sections by their discriminants. Degenerate conic sections. Sections of cones and eccentricity. 11.3 Polar coordinates Polar coordinates. Curves in polar coordinates. Polar equations of conic sections. Finding areas with polar coordinates. 11.R Review exercises

Chapter 12. Vectors 12.1 Vectors in the plane Vectors in a coordinate plane. Adding vectors and multiplying vectors by numbers. Displacement and position vectors. The unit vectors i and j. Force vectors. 12.2 The dot product in the plane The dot product and angles between vectors. Perpendicular vectors. The component of one vector in the direction of another. The orthogonal (perpendicular) projection of a vector on a line.

‡Procedures and programs for plotting sequences on a variety of calculators will be available on the web page for the text. p.viii (3/24/07) Table of Contents

12.3 Vectors in space Rectangular coordinates in space. The Pythagorean Theorem in space. Vectors in space. The dot product in space. Components and orthogonal projections of vectors in space. The unit vectors i, j, and k. Direction cosines and direction angles. 12.4 The cross product The cross product of two vectors. The scalar triple product. Areas of parallelograms and volumes of parallelepipeds. 12.5 Equations of lines and planes Parametric equations of lines in space. Perpendicular lines. Equations of planes. The distance from a point to a plane. Skew lines. Intercept equations of planes. Distance from a point to a line in an xy-plane. 12.R Review exercises

Chapter 13. Vector-Valued Functions 13.1 Parametric equations of curves Parametric equations of curves. Circular motion. Parametric equations of ellipses. Reversing a curve’s orientation. Parametric equations of curves given in polar coordinates. 13.2 Velocity Vectors Representing curves with vector-valued functions. Limits of vector-valued functions. Derivatives and continuity of vector-valued functions. Velocity vectors and speed. Lengths of parametric curves. Compound motion. Using relative velolcity vectors. 13.3 Acceleration vectors and Newton’s law of motion Integrating vector-valued functions. Acceleration vectors. The vector form of Newton’s Law of motion. 13.4 Curvature and acceleration in the plane Unit tangent and normal vectors to plane curves. Curvature of plane curves. Tangential and normal components of acceleration in the plane. Finding the tangential and normal components of an acceleration vector. 13.5 Curves in space and planetary motion The Product Rule for dot and cross products. Normal vectors and curvature of space curves. Normal and tangential components of acceleration in space. Kepler’s Laws of planetary motion and Newton’s Law of gravity. Velocity and acceleration vectors in polar coordinates. 13.R Review exercises

Chapter 14. Derivatives with Two or More Variables 14.1 Functions of two variables ...... p. 270 The domain, range, and graph of z = f(x,y). Fixing x or y: vertical cross sections of graphs. Drawing graphs of functions. 14.2 Horizontal cross sections of graphs and level curves ...... p. 288 Horizontal cross sections of graphs. Level curves. Estimating function values from level curves. Topographical maps and other contour curves. 14.3 Partial derivatives with two variables ...... p. 301 Limits of functions with two variables. Continuity of functions with two variables. Partial derivatives. A geometric interpretation of partial derivatives. Estimating partial derivatives from tables. Estimating partial derivatives from level curves. Table of Contents (3/24/07) p.ix

14.4 Chain Rules with two variables ...... p. 317 Using the Chain Rule with one variable. The general Chain Rule with two variables. Higher-order partial derivatives. 14.5 Directional derivatives and gradient vectors ...... p. 327 Directional derivatives. Using angles of inclination. Estimating directional derivatives from level curves. The gradient vector. Gradient vectors and level curves. Estimating gradient vectors from level curves. 14.6 Tangent planes and differentials ...... p. 342 Linear functions of two variables. Level curves of linear functions. Zooming in on level curves of a nonlinear z = f(x,y). Equations of tangent planes. Normal vectors. Differentials and error estimates. 14.7 Functions of three variables ...... p. 354 Functions of three variables and their level curves. Quadric surfaces. Partial derivatives. Gradient vectors and directional derivatives. Tangent planes to level surfaces. 14.8 Functions of more than three variables ...... p. 366 n-dimensional Euclidean space and functions of n variables. Partial derivatives. Graphs of functions and their tangent planes. Gradient vectors and directional derivatives. Level surfaces and their tangent planes. 14.R Review exercises ...... p. 372

Chapter 15. Maxima and Minima with Two or More Variables 15.1 The First-Derivative Test with two or three variables Local maxima and minima with two or three variables. The First-Derivative Test with two and three variables. Narrative problems. 15.2 The Second-Derivative Test with two variables The Second-Derivative Test and Taylor polynomials with one variable. Second-degree Taylor polynomials with two variables. The Second-Derivative Test with two variables. 15.3 Lagrange multipliers n Maxima and minima on curves in the plane and on surfaces in with n 3. < ≥ 15.R Review exercises

Chapter 16. Multiple Integrals 16.1 Double integrals Double integrals. Double integrals as volumes. Iterated integrals. Reversing the order of integration. ∆x∆y-notation in Riemann sums. 16.2 Applications of double integrals Volumes of regions between graphs z = f(x,y) and z = g(x,y). Two-dimensional density. Weights and centers of gravity of plates. Average value of z = f(x,y). 16.3 Double integrals in polar coordinates Double integrals in polar coordinates. Polar curves. Integration techniques (review). 16.4 Triple integrals Triple integrals in rectangular coordinates. Three-dimensional density and weight. Centers of gravity. Average value of w = f(x,y,z). p.x (3/24/07) Table of Contents

16.5 Triple integrals in cylindrical and spherical coordinates Cylindrical coordinates. Triple integrals in cylindrical coordinates. Spherical coordinates. Triple integrals in spherical cordinates. Spherical coordinates and geography. 16.6 Other changes of variables: Jacobians Affine transformations in the plane. Jacobians with two variables. Inverses of affine transformations in the plane. Affine changes of variables in double integrals. General changes of variables in double integrals. Jacobians and changes of variables in triple integrals. 16.R Review exercises

Chapter 17. Vector Analysis 17.1 Vector fields Vector fields in the plane and in space. Velocity fields and flow lines of a fluid in a plane. A rotational flow. Gradient vector fields. Force fields. 17.2 Line integrals Line integrals in the plane and in space. Interpreting line integrals with respect to arclength. Relating dx,dy, and dz to ds. Negatives, sums and differences of curves. Average value of a function on a curve. 17.3 Work and flux Work done by a constant force field along a line segment. Work done by a variable force field along a curve. Unit normal vectors to oriented plane curves. Flux of a constant velocity field across a line segment. Flux of a variable velocity field across an oriented curve. 17.4 Path independent line integrals Path independent line integrals and conservative force fields in two and three dimensions. 17.5 The Divergence and Stokes’ Theorems in the plane Green’s Theorem in the plane. Finding areas with line integrals. Circulation, curl, and Stokes’ Theorem in the plane. Divergence and the Divergence Theorem in the plane. Characterizing gradient fields in a plane (conclusion). Irrotational and incompressible fluid flow in the plane. Gradient fields. 17.6 Surface integrals Surface integrals. Average values on surfaces. Weights and centers of gravity of surfaces. 17.7 The Divergence and Stokes’ Theorems in space Gauss’ Theorem. Divergence and the Divergence Theorem in space. The curl and Stokes’ Theorem in space. Characterizing gradient fields in space (conclusion). 17.R Review exercises