CALCULUS SEVENTH EDITION. Indiana Academic Standards for Calculus. correlated to the CC2

Transcription

1 CALCULUS SEVENTH EDITION correlated to the Indiana Academic Standards for Calculus CC2 6/

2 Introduction to Calculus, 7 th Edition 2002 by Roland E. Larson, Robert P. Hostetler, Bruce H. Edwards Calculus, Seventh Edition is a college-level program that is appropriate for Advanced Placement students in high school. The text has comprehensive coverage of both single- and multivariable topics, making it appropriate for Advanced Placement Calculus BC. The Seventh Edition addresses the latest Advanced Placement guidelines by integrating the use of technology and emphasizing real-life data, practical applications, and mathematical models. The approach of Calculus, Seventh Edition is like that of a traditional text in that it is rooted in carefully developed theory, correct statements of theorems, inclusion of proofs, and mastery of traditional calculus skills. In addition, it embraces reform through technology, emphasis on reallife applications, conceptual and multi-part exercises, new explorations, and a myriad of student and teacher aids. Special Features P.S. Problem Solving a set of challenging exercises at the end of each chapter. These interesting problems not only draw upon and extend the chapter concepts, but they also allude to concepts that will be discussed in subsequent chapters. Getting at the Concept a set of exercises within each section that check students understanding of the basic concepts of the section. Think About It conceptual exercises that require students to use their critical-thinking skills and develop an intuitive understanding of the underlying theory of the calculus. Modeling Data multi-part questions that ask students to find and interpret mathematical models from real-life data, often through the use of a graphing utility. Writing in an effort to develop students reasoning skills and make them comfortable with discussing mathematical concepts, the text contains many writing exercises. Review Exercises these exercises are grouped and correlated by section, providing students with a way to focus on topics requiring review. Each chapter begins with a full-page chapter motivator that presents real-life situations with exploratory questions. As students attempt to use the techniques of their current skill set to answer the questions, they learn to appreciate the new calculus techniques presented in the chapter. Students finishing this course will be adequately prepared to take the Advanced Placement Examination in Calculus BC in the spring of their year of study. A complete listing of program components is provided on the following page.

3 Calculus, 7 th Edition 2002 Components Pupil s Edition Ancillaries let Test Item File, Volume 1 (Chapters P-10), Volume 2 (Chapters 10-14) Complete Solutions, Volume 1 (Chapters P-5) Complete Solutions, Volume 2 (Chapters 6-10) Complete Solutions, Volume 3 (Chapters 10-14) Graphing Test Generator HM ClassPrep Instructor s CD-ROM includes test items, lab manuals, AP Themes, and solutions to accompany the material in each chapter Graphing Calculator Video Calculus Video Program Calculus DVD Program Calculus Learning Tools Student CD-ROM includes explorations, labs, animations and other tools to support the material in the text Interactive Calculus 3.0 CD-ROM (entire book on CD-ROM) Internet Calculus 3.0 Web site access (entire book on web) Textbook web site

4 Calculus, Seventh Edition 2002 correlated to STANDARD 1 Limits and Continuity Students understand the concept of limit, find limits of functions at points and at infinity, decide if a function is continuous, and use continuity theorems. C.1.1 Understand the concept of limit and estimate limits from graphs and tables of values. Example: Estimate by calculating the function s values for x = 2.1, 2.01, and for x = 1.9, 1.99, C.1.2 Find limits by substitution. Example: Find C.1.3 Find limits of sums, differences, products, and quotients. Example: Find (sin x cos x + tan x). C.1.4 Find limits of rational functions that are undefined at a point. Example: Find by factoring and canceling Finding Limits Graphically and Numerically, Evaluating Limits Analytically, Evaluating Limits Analytically, Evaluating Limits Analytically, , 65, 66, 88, , 88, , 88, , 88, Volume I: 27-31, Section Volume I: 31-36, Section Volume I: 31-36, Section Volume I: 31-36, Section 1.3 = Pupil s Edition 1

5 C.1.5 Find one-sided limits. Example: Find C.1.6 Find limits at infinity. Example: Find C.1.7 Decide when a limit is infinite and use limits involving infinity to describe asymptotic behavior. Example: Find C.1.8 Find special limits such as. Example: Use a diagram to show that the limit above is equal to Continuity and One- Sided Limits, Limits at Infinity, Theme 2: Limits of Functions and Unbounded Behavior, Infinite Limits, 80-84; 3.5 Limits at Infinity, Theme 2: Limits of Functions and Unbounded Behavior, Evaluating Limits Analytically, , 89, , 236, , 89, 90-91, , 236, , 88, Volume I: 37-42, Section Volume I: , Section , Volume I: 42-46, 47-51, , Sections 1.5, Volume I: 31-36, Section 1.3 = Pupil s Edition 2

6 C.1.9 Understand continuity in terms of limits. Example: Show that f(x) = 3x + 1 is continuous at x = 2 by finding (3x + 1) and comparing it with f(2). 1.4 Continuity and One- Sided Limits, 68-76; 12.2 Limits and Continuity, Theme 2: Limits of Functions and Unbounded Behavior, , 89, 90-91, , , Volume I: 37-42, 47-51; Volume II: 80-83, Sections 1.4, 12.2 C.1.10 Decide if a function is continuous at a point. Example: Show that f(x) = is continuous at x = 2, provided that you define f(2) = Continuity and One- Sided Limits, 68-76; 12.2 Limits and Continuity, Theme 2: Limits of Functions and Unbounded Behavior, , 89, 90-91, , , Volume I: 37-42, 47-51; Volume II: 80-83, Sections 1.4, , Volume I: 37-42, 47-51; Volume II: 80-83, Sections 1.4, 7.8, 12.2 C.1.11 Find the types of discontinuities of a function. Example: What types of discontinuities has h(x) =? Explain your answer. 1.4 Continuity and One- Sided Limits, 69-74; 7.8 Infinite Discontinuity, 540; 12.2 Limits and Continuity, Theme 2: Limits of Functions and Unbounded Behavior, , 79, 89, 90-91, , 929 = Pupil s Edition 3

7 C.1.12 Understand and use the Intermediate Value Theorem on a function over a closed interval. Example: Show that g(x) = 3 x 2 has a zero between x = 1 and x = 2, because it is continuous. C.1.13 Understand and apply the Extreme Value Theorem: If f(x) is continuous over a closed interval, then f has a maximum and a minimum on the interval. Example: Decide if t(x) = tan x has a maximum value over the interval. What about the interval [-π, π]? Explain your answers. 1.4 Continuity and One- Sided Limits, Extrema on an Interval, ; 3.2 Rolle s Theorem, ; 12.8 Absolute Extrema and Relative Extrema, , , 235, Volume I: 37-42, Section , Volume I: , ; Volume II: , Sections 3.1, 3.2, 12.8 = Pupil s Edition 4

8 STANDARD 2 Differential Calculus Students find derivatives of algebraic, trigonometric, logarithmic, and exponential functions. They find derivatives of sums, products, and quotients, and composite and inverse functions. They find derivatives of higher order, and use logarithmic differentiation and the Mean Value Theorem. C.2.1 Understand the concept of derivative geometrically, numerically, and analytically, and interpret the derivative as a rate of change. Example: Find the derivative of f(x) = x 2 at x = 5 by calculating values of for x near 5. Use a diagram to explain what you are doing and what the result means. C.2.2 State, understand, and apply the definition of derivative. Example: Find What does the result tell you? 2.1 The Derivative and the Tangent Line Problem, ; 2.2 Rates of Change, ; 12.3 Partial Derivatives, Theme 3: The Derivative at a Point and the Derivative as A Function, The Derivative of a Function, 97-99; 12.3 Partial Derivatives, Theme 3: The Derivative at a Point and the Derivative as A Function, , , 153, , 153, , , Volume I: 53-67, 92-97; Volume II: 83-88, Sections 2.1, 2.2, , Volume I: 53-60, 92-97; Volume II: 83-88, Sections 2.1, 12.3 = Pupil s Edition 5

9 C.2.3 Find the derivatives of functions, including algebraic, trigonometric, logarithmic, and exponential functions. Example: Find for the function y = x 5. C.2.4 Find the derivatives of sums, products, and quotients. Example: Find the derivative of x cos x.= 2.1 The Derivative of a Function, 97-99; 2.2 Basic Differentiation Rules, ; Derivatives of Sine and Cosine Functions, 110; 2.3 Derivatives of Trigonometric Functions, ; 2.4 Trigonometric Functions and the Chain Rule, 132; 5.1 The Derivative of the Natural Logarithmic Function, ; 5.4 Derivatives of Exponential Functions, ; 5.5 Derivatives for Bases Other than e, 353, 354; 5.10 Derivatives of Hyperbolic Functions, ; 12.3 Partial Derivatives, ; 12.5 Chain Rules for Functions of Several Variables, Theme 3: The Derivative at a Point and the Derivative as A Function, The Sum and Difference Rules, 109; 2.3 The Product and Quotient Rules, ; 2.4 Simplifying Derivatives of Products and Quotients, 131 Theme 3: The Derivative at a Point and the Derivative as A Function, , , , , , , 348, 357, 403, , , , , , , , , , Volume I: 53-79, 92-97, , , ; Volume II: 83-88, Sections , 5.1, 5.4, 5.5, 5.10, 12.3, : Volume I: 60-79, Sections = Pupil s Edition 6

10 C.2.5 Find the derivatives of composite functions, using the chain rule. Example: Find f (x) for f(x) = (x 2 +2) The Chain Rule, ; 12.5 Chain Rules for Functions of Several Variables, Theme 3: The Derivative at a Point and the Derivative as A Function, , , , , Volume I: 72-79, 92-97; Volume II: 92-97, Sections 2.4, 12.5 C.2.6 Find the derivatives of implicitly-defined functions. Example: For xy x 2 y 2 = 5, find at the point (2, 3). 2.5 Implicit Differentiation, ; 12.5 Implicit Partial Differentiation, , 155, , , Volume I: 79-85, 92-97; Volume II: 92-97, Sections 2.4, 12.5 C.2.7 Find derivatives of inverse functions. Example: Let f(x) = 2x 3 and g = f -1. Find g (2). 5.3 Derivative of an Inverse Function, ; 5.8 Derivatives of Inverse Trigonometric Functions, ; 5.10 Differentiation of Inverse Hyperbolic Functions, Theme 9: Functions and Their Inverses, , 340, , 404, Volume I: , , Sections 5.3, 5.8, 5.10 C.2.8 Find second derivatives and derivatives of higher order. Example: Find the second derivative of e 5x. 2.3 Higher Order Derivatives, 123; 12.3 Higher-Order Partial Derivatives, , , , , Volume I: 67-73, 92-97; Volume II: 83-88, Sections 2.3, 12.3 = Pupil s Edition 7

11 C.2.9 Find derivatives using logarithmic differentiation. Example: Find for y =. 5.1 Logarithmic Differentiation, ; 5.5 Differentiation to Other Bases, 353, , 357, 358, Volume I: , , Sections 5.1, 5.5 C.2.10 Understand and use the relationship between differentiability and continuity. Example: Is f(x) = continuous at x = 0? Is f(x) differentiable at x 0? Explain your answers. 2.1 Differentiability and Continuity, ; 5.3 Continuity and Differentiability of Inverse Functions, ; 12.4 Differentiability, ; Appendix B Continuity and Differentiability of Inverse Functions, A19 Theme 2: Limits of Functions and Unbounded Behavior, , 153, 340, , , Theme 2: Sample 7-8 Volume I: 53-60; Volume II: 89 Sections 2.1, 5.3, 12.4 C.2.11 Understand and apply the Mean Value Theorem. Example: For f(x) = on the interval [1, 9], find the value of c such that 3.2 The Mean Value Theorem, ; Appendix B The Extended Mean Value Theorem, A21- A , 235, 238, Volume I: , Section 3.2 = Pupil s Edition 8

12 STANDARD 3 Applications of Derivatives Students find slopes and tangents, maximum and minimum points, and points of inflection. They solve optimization problems and find rates of change. C.3.1 Find the slope of a curve at a point, including points at which there are vertical tangents and no tangents. Example: Find the slope of the tangent to y = x 3 at the point (2, 8). C.3.2 Find a tangent line to a curve at a point and a local linear approximation. Example: In the last example, find an equation of the tangent at (2, 8). C.3.3 Decide where functions are decreasing and increasing. Understand the relationship between the increasing and decreasing behavior of f and the sign of f. Example: Use values of the derivative to find where f(x) = x 3 3x is decreasing. 2.1 The Derivative and the Tangent Line Problem, 94-99, The Derivative and the Tangent Line Problem, 96-99; 3.8 Newton s Method, ; 3.9 Differentials, Increasing and Decreasing Functions and the First Derivative Test, Theme 4: The Graphical Relationship Between First and Second Derivatives, , 153, , 153, 156, , , , 235, Volume I: 53-60, Section , Volume I: 53-60, , Sections 2.1, 3.8, Volume I: , Section 3.3 = Pupil s Edition 9

13 C.3.4 Find local and absolute maximum and minimum points. Example: In the last example, find the local maximum and minimum points of f(x). 3.1 Extrema on an Interval, ; 3.3 The First Derivative Test, ; 5.1 Natural Log: Relative Extrema, 320; 5.4 Exponential Functions: Relative Extrema, 343; 5.10 Hyperbolic Functions: Relative Extrema, 398; 12.8 Extrema of Functions of Two Variables, Theme 4: The Graphical Relationship Between First and Second Derivatives, , , 235, , 322, 323, 348, 403, 406, 408, , , , Volume I: , , , , , ; Volume II: , Sections 3.1, 3.3, 5.1, 5.4, C.3.5 Analyze curves, including the notions of monotonicity and concavity. Example: In the last example, for which values of x is f(x) decreasing and for which values of x is f(x) concave upward? 3.3 Increasing and Decreasing Functions and the First Derivative Test, ; 3.4 Concavity and the Second Derivative Test, Theme 4: The Graphical Relationship Between First and Second Derivatives, , , , Volume I: , Sections 3.3, 3.4 C.3.6 Find points of inflection of functions. Understand the relationship between the concavity of f and the sign of f. Understand points of inflection as places where concavity changes. Example: In the last example, find the points of inflection of f(x) and where f(x) is concave upward and concave downward. 3.4 Concavity and the Second Derivative Test, Theme 4: The Graphical Relationship Between First and Second Derivatives, , 236, Volume I: , Section 3.4 = Pupil s Edition 10

14 C.3.7 Use first and second derivatives to help sketch graphs. Compare the corresponding characteristics of the graphs of f, f, and f. Example: Use the last examples to draw the graph of f(x) = x 3 3x. 3.6 A Summary of Curve Sketching, ; 12.3 Partial Derivatives and Surfaces, Theme 4: The Graphical Relationship Between First and Second Derivatives, , , , 866, 929 : 66-77, Volume I: , ; Volume II: 84-85, 124 Sections 3.6, 12.3 C.3.8 Use implicit differentiation to find the derivative of an inverse function. Example: Let f(x) = 2x 3 and g = f -1. Find g (x) using implicit differentiation. 5.3 Derivative of an Inverse Function, ; 5.8 Derivatives of Inverse Trigonometric Functions, ; 5.10 Differentiation of Inverse Hyperbolic Functions, , 340, , 404, Volume I: , , 273, 277 Sections 5.3, 5.8, 5.10 C.3.9 Solve optimization problems. Example: You want to enclose a rectangular area of 5,000 m 2. Find the shortest length of fencing you can use. 3.7 Optimization Problems, ; 12.9 Applications of Extrema of Functions of Two Variables, ; Lagrange Multipliers, , 237, , , , 931, , Volume I: , , 175; Volume II: , , Sections 3.7, 12.9, = Pupil s Edition 11

15 C.3.10 Find average and instantaneous rates of change. Understand the instantaneous rate of change as the limit of the average rate of change. Interpret a derivative as a rate of change in applications, including velocity, speed, and acceleration. Example: You are filling a bucket with water and the height H cm of the water after t seconds is given by 2.2 Rates of Changes, ; 2.3 Acceleration Due to Gravity, 123; 3.2 Instantaneous Rate of Change, 171 Theme 3: The Derivative at a Point and the Derivative as A Function, , 125, 126, , 157, 172, 173, Volume I: 60-67, 71-73, 93-95, 101, , Sections 2.2, 2.3, 3.2. How fast is the water rising 30 seconds after you start filling the bucket? Explain your answer. C.3.11 Find the velocity and acceleration of a particle moving in a straight line. Example: A bead on a wire moves so that, after t seconds, its distance s cm from the midpoint of the wire is given by Find its maximum velocity and where along the wire this occurs. 2.2 Rates of Changes, ; 2.3 Acceleration Due to Gravity, 123; 3.2 Instantaneous Rate of Change, , 125, 126, , 157, 172, 173, Volume I: 60-67, 71-73, 93-95, 101, , Sections 2.2, 2.3, 3.2 C.3.12 Model rates of change, including related rates problems. Example: A boat is heading south at 10 mph. Another boat is heading west at 15 mph toward the same point. At these speeds, they will collide. Find the rate that the distance between them is decreasing 1 hour before they collide. 2.2 Rates of Changes, ; 2.6 Related Rates, , 12.5 Related Rates, 877 Theme 3: The Derivative at a Point and the Derivative as A Function, , , , 155, 157, 877, , Volume I: 64-67, 85-92, 94, 97, ; Volume II: 96, 130 Sections 2.2, 2.6, 12.5 = Pupil s Edition 12

16 STANDARD 4 Integral Calculus Students define integrals using Riemann Sums, use the Fundamental Theorem of Calculus to find integrals, and use basic properties of integrals. They integrate by substitution and find approximate integrals. C.4.1 Use rectangle approximations to find approximate values of integrals. Example: Find an approximate value for using 6 rectangles of equal width under the graph of f(x) = x 2. 4.A Wankel Engine and Area, 240; 4.2 Area, , , Volume I: , Section 4.2 C.4.2 Calculate the values of Riemann Sums over equal subdivisions using left, right, and midpoint evaluation points. Example: Find the value of the Riemann Sum over the interval [0, 3] using 6 subintervals of equal width for f(x) = x 2 evaluated at the midpoint of each subinterval. 4.2 Area, , Volume I: , Section 4.2 C.4.3 Interpret a definite integral as a limit of Riemann Sums. Example: Find the values of the Riemann Sums over the interval [0, 3] using 12, 24, etc., subintervals of equal width for f(x) = x 2 evaluated at the midpoint of each subinterval. Find the limit of the Riemann Sums. 4.3 Reimann Sums and Definite Integrals, Theme 5: The Definite Integral as Total Change, , Volume I: , 211 Section 4.3 = Pupil s Edition 13

17 C.4.4 Understand the Fundamental Theorem of Calculus: Interpret a definite integral of the rate of change of a quantity over an interval as the change of the quantity over the interval, that is Example: Explain why. 4.4 The Fundamental Theorem of Calculus, Theme 5: The Definite Integral as Total Change, , Volume I: , Section 4.4 C.4.5 Use the Fundamental Theorem of Calculus to evaluate definite and indefinite integrals and to represent particular antiderivatives. Perform analytical and graphical analysis of functions so defined. Example: Using antiderivatives, find 4.4 The Fundamental Theorem of Calculus, , Volume I: , Section 4.4 C.4.6 Understand and use these properties of definite integrals: 4.3 Properties of Definite Integrals, , 308, Volume I: , 211, Section 4.3 = Pupil s Edition 14

18 C.4.7 Understand and use integration by substitution (or change of variable) to find values of integrals. Example: Find 4.5 Integration by Substitution, ; 7.4 Trigonometric Substitution, , 308, , , , Volume I: , , , Sections 4.5, 7.4 C.4.8 Understand and use Riemann Sums, the Trapezoidal Rule, and technology to approximate definite integrals of functions represented algebraically, geometrically, and by tables of values. Example: Use the Trapezoidal Rule with 6 subintervals over [0, 3] for f(x) = x 2 to approximate the value of 4.6 Numerical Integration, Theme 5: The Definite Integral as Total Change, , 309, Volume I: , Section 4.6 = Pupil s Edition 15

19 STANDARD 5 Applications of Integration Students find velocity functions and position functions from their derivatives, solve separable differential equations, and use definite integrals to find areas and volumes. C.5.1 Find specific antiderivatives using initial conditions, including finding velocity functions from acceleration functions, finding position functions from velocity functions, and applications to motion along a line. Example: A bead on a wire moves so that its velocity, after t seconds, is given by v(t) = 3 cos 3t. Given that it starts 2 cm to the left of the midpoint of the wire, find its position after 5 seconds. C.5.2 Solve separable differential equations and use them in modeling. Example: The slope of the tangent to the curve y = f(x) is given by. Find an equation of the curve y = f(x). 4.1 Antiderivatives and Indefinite Integration, ; 4.5 Integration by Substitution, ; 5.2 The Natural Logarithmic Function: Integration, ; 5.4 Integrals of Exponential Functions, ; 5.5 Bases Other than e, 354; 5.9 Inverse Trigonometric Functions: Integration, ; 5.10 Integration of Hyperbolic and Inverse Hyperbolic Functions, 397, 399, 401, 402 Theme 8: Rectilinear Motion, Differential Equations: Separation of Variables, Theme 10: Writing and Solving Separable Differential Equations and Modeling, 37-38; Theme 11: Differential Equations, Slope Field, and the Logistics Equation, , , , 310, , 349, 357, , , , 407, , Volume I: , , 209, 212, , , , , , , 274, 275, 277 Sections 4.1, 4.5, 5.2, 5.4, 5.5, 5.9, , Volume I: , , Section 5.7 = Pupil s Edition 16

20 C.5.3 Solve differential equations of the form y = ky as applied to growth and decay problems. Example: The amount of a certain radioactive material was 10 kg a year ago. The amount is now 9 kg. When will it be reduced to 1 kg? Explain your answer. 5.6 Differential Equations: Growth and Decay, Theme 10: Writing and Solving Separable Differential Equations and Modeling, 37-38; Theme 11: Differential Equations, Slope Field, and the Logistics Equation, , 407, , Volume I: , 275, Section 5.6 C.5.4 Use definite integrals to find the area between a curve and the x-axis, or between two curves. Example: Find the area bounded by and x = 2. 6.A Constructing an Arch Dam, 410; 6.1 Area of a Region Between Two Curves, ; 13.1 Iterated Integrals and Area in the Plane, , , 476, , , 1001, , Volume I: , , ; Volume II: , 169, 174 Sections 6.1, 13.1 C.5.5 Use definite integrals to find the average value of a function over a closed interval. Example: Find the average value of over [0, 2]. 4.4 Average Value of a Function, , Volume I: , 212 Section 4.4 = Pupil s Edition 17

21 C.5.6 Use definite integrals to find the volume of a solid with known cross-sectional area. Example: A cone with its vertex at the origin lies symmetrically along the x- axis. The base of the cone is at x = 5 and the base radius is 7. Use integration to find the volume of the cone. 6.A Constructing an Arch Dam, 410; 6.2 Volume: The Disk Method, ; 6.3 Volume: The Shell Method, ; 13.2 Double Integrals and Volume, ; 13.3 Polar Coordinates and Volume, ; 13.6 Triple Integrals and Volume, ; 13.7 Cylindrical and Spherical Coordinates and Volume, , 992 Theme 7: Volumes with Know Cross Sections and Other Applications of Integration, , , , , 478, , , , , , , Volume I: , , ; Volume II: , 158, , , Sections 6.2, 6.3, 13.2, 13.3, 13.6, 13.7 C.5.7 Apply integration to model and solve problems in physics, biology, economics, etc., using the integral as a rate of change to give accumulated change and using the method of setting up an approximating Riemann Sum and representing its limit as a definite integral. Example: Find the amount of work done by a variable force. 4.4 Applications of Fundamental Theorem, 280; 6.4 Arc Length and Surfaces of Revolution, ; 6.5 Work, ; 6.6 Moments, Centers of Mass, and Centroids, ; 6.7 Fluid Pressure and Fluid Force, ; 13.4 Center of Mass and Moments of Inertia, ; 13.5 Surface Area, ; 13.6 Triple Integrals and Applications, ; 13.7 Triple Integrals in Cylindrical and Spherical Coordinates, 990, 992 Theme 5: The Definite Integral as Total Change, 17-18; Theme 6: The Integral as an Accumulation Function, , 299, 306, 309, 311, 331, 339, 350, , 409, , , , , , 479, 970, 977, , , , , , , , Volume I: , , 208, 213, 216, 226, 229, 239, , , ; Volume II: , , , , 176 Sections , , ; see also Appendix G: Business and Economic Applications = Pupil s Edition 18