Correlations

Calculus with Applications, Brief, 8th Edition ©2005

Lial

Calculus AB

Correlated to: Advanced Placement* (AP*) Calculus AB Standards (Grades 9–12)

SE = Student Edition
TE = Teacher Edition
TR = Teaching Resource
TECH = Technology

I. FUNCTIONS, GRAPHS, AND LIMITS

Analysis of graphs. With the aid of technology, graphs of functions are often easy to produce. The emphasis is on the interplay between the geometric and analytic information and on the use of calculus both to predict and to explain the observed local and global behavior of a function.
Limits of functions (including one-sided limits)
—An intuitive understanding of the limiting process.	SE/TE: 140–158, 485 TR: Instructor's Resource Guide and Solutions Manual provides solutions, tests, exams, and teaching tips; Videotape Series provides coverage of most topics; TestGen-EQ; PowerPoint Lecture Presentation TECH: Graphing Calculator and Excel Spreadsheet Manual; Digital Video Tutor; Addison-Wesley Math Tutor Center (www.aw-bc.com/tutorcenter); MathXL; MyMathLab; InterActMath (www.interactmath.com)
—Calculating limits using algebra.	SE/TE: 140–158 TR: Instructor's Resource Guide and Solutions Manual provides solutions, tests, exams, and teaching tips; Videotape Series provides coverage of most topics; TestGen-EQ; PowerPoint Lecture Presentation TECH: Graphing Calculator and Excel Spreadsheet Manual; Digital Video Tutor; Addison-Wesley Math Tutor Center (www.aw-bc.com/tutorcenter); MathXL; MyMathLab; InterActMath (www.interactmath.com)
—Estimating limits from graphs or tables of data.	SE/TE: 142, 143, 148, 149, 150, 153, 154, 156–157 TR: Instructor's Resource Guide and Solutions Manual provides solutions, tests, exams, and teaching tips; Videotape Series provides coverage of most topics; TestGen-EQ; PowerPoint Lecture Presentation TECH: Graphing Calculator and Excel Spreadsheet Manual; Digital Video Tutor; Addison-Wesley Math Tutor Center (www.aw-bc.com/tutorcenter); MathXL; MyMathLab; InterActMath (www.interactmath.com)
Asymptotic and unbounded behavior
—Understanding asymptotes in terms of graphical behavior	SE/TE: 81–83, 85–86, 314–315 TR: Instructor's Resource Guide and Solutions Manual provides solutions, tests, exams, and teaching tips; Videotape Series provides coverage of most topics; TestGen-EQ; PowerPoint Lecture Presentation TECH: Graphing Calculator and Excel Spreadsheet Manual; Digital Video Tutor; Addison-Wesley Math Tutor Center (www.aw-bc.com/tutorcenter); MathXL; MyMathLab; InterActMath (www.interactmath.com)
—Describing asymptotic behavior in terms of limits involving infinity	SE/TE: 150 TR: Instructor's Resource Guide and Solutions Manual provides solutions, tests, exams, and teaching tips; Videotape Series provides coverage of most topics; TestGen-EQ; PowerPoint Lecture Presentation TECH: Graphing Calculator and Excel Spreadsheet Manual; Digital Video Tutor; Addison-Wesley Math Tutor Center (www.aw-bc.com/tutorcenter); MathXL; MyMathLab; InterActMath (www.interactmath.com)
—Comparing relative magnitudes of functions and their rates of change. (For example, contrasting exponential growth, polynomial growth, and logarithmic growth)	SE/TE: 129, 130–131 TR: Instructor's Resource Guide and Solutions Manual provides solutions, tests, exams, and teaching tips; Videotape Series provides coverage of most topics; TestGen-EQ; PowerPoint Lecture Presentation TECH: Graphing Calculator and Excel Spreadsheet Manual; Digital Video Tutor; Addison-Wesley Math Tutor Center (www.aw-bc.com/tutorcenter); MathXL; MyMathLab; InterActMath (www.interactmath.com)
Continuity as a property of functions
—An intuitive understanding of continuity. (Close values of the domain lead to close values of the range.)	SE/TE: 158–166 TR: Instructor's Resource Guide and Solutions Manual provides solutions, tests, exams, and teaching tips; Videotape Series provides coverage of most topics; TestGen-EQ; PowerPoint Lecture Presentation TECH: Graphing Calculator and Excel Spreadsheet Manual; Digital Video Tutor; Addison-Wesley Math Tutor Center (www.aw-bc.com/tutorcenter); MathXL; MyMathLab; InterActMath (www.interactmath.com)
—Understanding continuity in terms of limits	SE/TE: 159, 161–166 TR: Instructor's Resource Guide and Solutions Manual provides solutions, tests, exams, and teaching tips; Videotape Series provides coverage of most topics; TestGen-EQ; PowerPoint Lecture Presentation TECH: Graphing Calculator and Excel Spreadsheet Manual; Digital Video Tutor; Addison-Wesley Math Tutor Center (www.aw-bc.com/tutorcenter); MathXL; MyMathLab; InterActMath (www.interactmath.com)
—Geometric understanding of graphs of continuous functions (Intermediate Value Theorem and Extreme Value Theorem)	SE/TE: 159–166, 326–335, 339 TR: Instructor's Resource Guide and Solutions Manual provides solutions, tests, exams, and teaching tips; Videotape Series provides coverage of most topics; TestGen-EQ; PowerPoint Lecture Presentation TECH: Graphing Calculator and Excel Spreadsheet Manual; Digital Video Tutor; Addison-Wesley Math Tutor Center (www.aw-bc.com/tutorcenter); MathXL; MyMathLab; InterActMath (www.interactmath.com)

II. DERIVATIVES

Concept of the derivative
—Derivative presented graphically, numerically, and analytically	SE/TE: 178–196, 197–204, 211–228, 228–236, 236–247, 247–255, 255–263, 269–282, 282–295, 295–310, 310–320, 326–335, 335–346, 346–355, 355–361, 362–368, 369–375 TR: Instructor's Resource Guide and Solutions Manual provides solutions, tests, exams, and teaching tips; Videotape Series provides coverage of most topics; TestGen-EQ; PowerPoint Lecture Presentation TECH: Graphing Calculator and Excel Spreadsheet Manual; Digital Video Tutor; Addison-Wesley Math Tutor Center (www.aw-bc.com/tutorcenter); MathXL; MyMathLab; InterActMath (www.interactmath.com)
—Derivative interpreted as an instantaneous rate of change	SE/TE: 169–178, 183 TR: Instructor's Resource Guide and Solutions Manual provides solutions, tests, exams, and teaching tips; Videotape Series provides coverage of most topics; TestGen-EQ; PowerPoint Lecture Presentation TECH: Graphing Calculator and Excel Spreadsheet Manual; Digital Video Tutor; Addison-Wesley Math Tutor Center (www.aw-bc.com/tutorcenter); MathXL; MyMathLab; InterActMath (www.interactmath.com)
—Derivative defined as the limit of the different quotient	SE/TE: 167, 183 TR: Instructor's Resource Guide and Solutions Manual provides solutions, tests, exams, and teaching tips; Videotape Series provides coverage of most topics; TestGen-EQ; PowerPoint Lecture Presentation TECH: Graphing Calculator and Excel Spreadsheet Manual; Digital Video Tutor; Addison-Wesley Math Tutor Center (www.aw-bc.com/tutorcenter); MathXL; MyMathLab; InterActMath (www.interactmath.com)
—Relationship between differentiability and continuity	SE/TE: 190–191 TR: Instructor's Resource Guide and Solutions Manual provides solutions, tests, exams, and teaching tips; Videotape Series provides coverage of most topics; TestGen-EQ; PowerPoint Lecture Presentation TECH: Graphing Calculator and Excel Spreadsheet Manual; Digital Video Tutor; Addison-Wesley Math Tutor Center (www.aw-bc.com/tutorcenter); MathXL; MyMathLab; InterActMath (www.interactmath.com)
Derivative at a point
—Slope of a curve at a point. Examples are emphasized, including points at which there are vertical tangents and points at which there are no tangents.	SE/TE: 180–182, 184–185, 191 TR: Instructor's Resource Guide and Solutions Manual provides solutions, tests, exams, and teaching tips; Videotape Series provides coverage of most topics; TestGen-EQ; PowerPoint Lecture Presentation TECH: Graphing Calculator and Excel Spreadsheet Manual; Digital Video Tutor; Addison-Wesley Math Tutor Center (www.aw-bc.com/tutorcenter); MathXL; MyMathLab; InterActMath (www.interactmath.com)
—Tangent line to a curve at a point and local linear approximation	SE/TE: 179–182, 300–301, 369–375 TR: Instructor's Resource Guide and Solutions Manual provides solutions, tests, exams, and teaching tips; Videotape Series provides coverage of most topics; TestGen-EQ; PowerPoint Lecture Presentation TECH: Graphing Calculator and Excel Spreadsheet Manual; Digital Video Tutor; Addison-Wesley Math Tutor Center (www.aw-bc.com/tutorcenter); MathXL; MyMathLab; InterActMath (www.interactmath.com)
—Instantaneous rate of change as the limit of average rate of change	SE/TE: 169–178 TR: Instructor's Resource Guide and Solutions Manual provides solutions, tests, exams, and teaching tips; Videotape Series provides coverage of most topics; TestGen-EQ; PowerPoint Lecture Presentation TECH: Graphing Calculator and Excel Spreadsheet Manual; Digital Video Tutor; Addison-Wesley Math Tutor Center (www.aw-bc.com/tutorcenter); MathXL; MyMathLab; InterActMath (www.interactmath.com)
—Approximate rate of change from graphs and tables of values	SE/TE: 174, 176, 177–178, 198 TR: Instructor's Resource Guide and Solutions Manual provides solutions, tests, exams, and teaching tips; Videotape Series provides coverage of most topics; TestGen-EQ; PowerPoint Lecture Presentation TECH: Graphing Calculator and Excel Spreadsheet Manual; Digital Video Tutor; Addison-Wesley Math Tutor Center (www.aw-bc.com/tutorcenter); MathXL; MyMathLab; InterActMath (www.interactmath.com)
Derivative as a function
—Corresponding characteristics of graphs of f and f′	SE/TE: 199–204 TR: Instructor's Resource Guide and Solutions Manual provides solutions, tests, exams, and teaching tips; Videotape Series provides coverage of most topics; TestGen-EQ; PowerPoint Lecture Presentation TECH: Graphing Calculator and Excel Spreadsheet Manual; Digital Video Tutor; Addison-Wesley Math Tutor Center (www.aw-bc.com/tutorcenter); MathXL; MyMathLab; InterActMath (www.interactmath.com)
—Relationship between the increasing and decreasing behavior of f and the sign of f′	SE/TE: 273–282 TR: Instructor's Resource Guide and Solutions Manual provides solutions, tests, exams, and teaching tips; Videotape Series provides coverage of most topics; TestGen-EQ; PowerPoint Lecture Presentation TECH: Graphing Calculator and Excel Spreadsheet Manual; Digital Video Tutor; Addison-Wesley Math Tutor Center (www.aw-bc.com/tutorcenter); MathXL; MyMathLab; InterActMath (www.interactmath.com)
—The Mean Value Theorem and its geometric consequences	not covered
—Equations involving derivatives. Verbal descriptions are translated into equations involving derivatives and vice versa	SE/TE: 199–210, 251, 335–346, 346–355, 362–368, 373–375 TR: Instructor's Resource Guide and Solutions Manual provides solutions, tests, exams, and teaching tips; Videotape Series provides coverage of most topics; TestGen-EQ; PowerPoint Lecture Presentation TECH: Graphing Calculator and Excel Spreadsheet Manual; Digital Video Tutor; Addison-Wesley Math Tutor Center (www.aw-bc.com/tutorcenter); MathXL; MyMathLab; InterActMath (www.interactmath.com)
Second derivatives
—Corresponding characteristics of the graphs of f, f′, and f″	SE/TE: 299–310 TR: Instructor's Resource Guide and Solutions Manual provides solutions, tests, exams, and teaching tips; Videotape Series provides coverage of most topics; TestGen-EQ; PowerPoint Lecture Presentation TECH: Graphing Calculator and Excel Spreadsheet Manual; Digital Video Tutor; Addison-Wesley Math Tutor Center (www.aw-bc.com/tutorcenter); MathXL; MyMathLab; InterActMath (www.interactmath.com)
—Relationship between the concavity of f and the sign of f″	SE/TE: 299–303 TR: Instructor's Resource Guide and Solutions Manual provides solutions, tests, exams, and teaching tips; Videotape Series provides coverage of most topics; TestGen-EQ; PowerPoint Lecture Presentation TECH: Graphing Calculator and Excel Spreadsheet Manual; Digital Video Tutor; Addison-Wesley Math Tutor Center (www.aw-bc.com/tutorcenter); MathXL; MyMathLab; InterActMath (www.interactmath.com)
—Points of inflection as places where concavity changes	SE/TE: 299–303 TR: Instructor's Resource Guide and Solutions Manual provides solutions, tests, exams, and teaching tips; Videotape Series provides coverage of most topics; TestGen-EQ; PowerPoint Lecture Presentation TECH: Graphing Calculator and Excel Spreadsheet Manual; Digital Video Tutor; Addison-Wesley Math Tutor Center (www.aw-bc.com/tutorcenter); MathXL; MyMathLab; InterActMath (www.interactmath.com)
Applications of derivatives
—Analysis of curves, including the notions of monotonicity and concavity	SE/TE: 310–318 TR: Instructor's Resource Guide and Solutions Manual provides solutions, tests, exams, and teaching tips; Videotape Series provides coverage of most topics; TestGen-EQ; PowerPoint Lecture Presentation TECH: Graphing Calculator and Excel Spreadsheet Manual; Digital Video Tutor; Addison-Wesley Math Tutor Center (www.aw-bc.com/tutorcenter); MathXL; MyMathLab; InterActMath (www.interactmath.com)
—Optimization, both absolute (global) and relative (local) extrema	SE/TE: 326–335, 335–346 TR: Instructor's Resource Guide and Solutions Manual provides solutions, tests, exams, and teaching tips; Videotape Series provides coverage of most topics; TestGen-EQ; PowerPoint Lecture Presentation TECH: Graphing Calculator and Excel Spreadsheet Manual; Digital Video Tutor; Addison-Wesley Math Tutor Center (www.aw-bc.com/tutorcenter); MathXL; MyMathLab; InterActMath (www.interactmath.com)
—Modeling rates of change, including related rates problems	SE/TE: 362–368 TR: Instructor's Resource Guide and Solutions Manual provides solutions, tests, exams, and teaching tips; Videotape Series provides coverage of most topics; TestGen-EQ; PowerPoint Lecture Presentation TECH: Graphing Calculator and Excel Spreadsheet Manual; Digital Video Tutor; Addison-Wesley Math Tutor Center (www.aw-bc.com/tutorcenter); MathXL; MyMathLab; InterActMath (www.interactmath.com)
—Use of implicit differentiation to find the derivative of an inverse function.	SE/TE: 355–361 TR: Instructor's Resource Guide and Solutions Manual provides solutions, tests, exams, and teaching tips; Videotape Series provides coverage of most topics; TestGen-EQ; PowerPoint Lecture Presentation TECH: Graphing Calculator and Excel Spreadsheet Manual; Digital Video Tutor; Addison-Wesley Math Tutor Center (www.aw-bc.com/tutorcenter); MathXL; MyMathLab; InterActMath (www.interactmath.com)
—Interpretation of the derivative as a rate of change in varied applied contexts, including velocity, speed, and acceleration	SE/TE: 170–173, 362–368 TR: Instructor's Resource Guide and Solutions Manual provides solutions, tests, exams, and teaching tips; Videotape Series provides coverage of most topics; TestGen-EQ; PowerPoint Lecture Presentation TECH: Graphing Calculator and Excel Spreadsheet Manual; Digital Video Tutor; Addison-Wesley Math Tutor Center (www.aw-bc.com/tutorcenter); MathXL; MyMathLab; InterActMath (www.interactmath.com)
—Geometric interpretation of differential equations via slope fields and the relationship between slope fields and solution curves for differential equations	not covered
Computation of derivatives
—Knowledge of derivatives of basic functions, including power, exponential, logarithmic, trigonometric, and inverse trigonometric functions	SE/TE: 214–215, 237–251, 255–260 TR: Instructor's Resource Guide and Solutions Manual provides solutions, tests, exams, and teaching tips; Videotape Series provides coverage of most topics; TestGen-EQ; PowerPoint Lecture Presentation TECH: Graphing Calculator and Excel Spreadsheet Manual; Digital Video Tutor; Addison-Wesley Math Tutor Center (www.aw-bc.com/tutorcenter); MathXL; MyMathLab; InterActMath (www.interactmath.com)
—Basic rules for the derivative of sums, products, and quotients of functions.	SE/TE: 214–227, 228–236, TR: Instructor's Resource Guide and Solutions Manual provides solutions, tests, exams, and teaching tips; Videotape Series provides coverage of most topics; TestGen-EQ; PowerPoint Lecture Presentation TECH: Graphing Calculator and Excel Spreadsheet Manual; Digital Video Tutor; Addison-Wesley Math Tutor Center (www.aw-bc.com/tutorcenter); MathXL; MyMathLab; InterActMath (www.interactmath.com)
—Chain rule and implicit differentiation	SE/TE: 236–247, 355–359 TR: Instructor's Resource Guide and Solutions Manual provides solutions, tests, exams, and teaching tips; Videotape Series provides coverage of most topics; TestGen-EQ; PowerPoint Lecture Presentation TECH: Graphing Calculator and Excel Spreadsheet Manual; Digital Video Tutor; Addison-Wesley Math Tutor Center (www.aw-bc.com/tutorcenter); MathXL; MyMathLab; InterActMath (www.interactmath.com)

III. INTEGRALS

Interpretations and properties of definite integrals
—Computation of Riemann sums using left, right, and midpoint evaluation points	SE/TE: 403–408 TR: Instructor's Resource Guide and Solutions Manual provides solutions, tests, exams, and teaching tips; Videotape Series provides coverage of most topics; TestGen-EQ; PowerPoint Lecture Presentation TECH: Graphing Calculator and Excel Spreadsheet Manual; Digital Video Tutor; Addison-Wesley Math Tutor Center (www.aw-bc.com/tutorcenter); MathXL; MyMathLab; InterActMath (www.interactmath.com)
—Definite integral as a limit of Riemann sums over equal subdivisions	SE/TE: 403–408 TR: Instructor's Resource Guide and Solutions Manual provides solutions, tests, exams, and teaching tips; Videotape Series provides coverage of most topics; TestGen-EQ; PowerPoint Lecture Presentation TECH: Graphing Calculator and Excel Spreadsheet Manual; Digital Video Tutor; Addison-Wesley Math Tutor Center (www.aw-bc.com/tutorcenter); MathXL; MyMathLab; InterActMath (www.interactmath.com)
—Definite integral of the rate of change of a quantity over an interval interpreted as the change of the quantity over the interval: f′(x)dx = f(b) – f(a)	SE/TE: 410–417 TR: Instructor's Resource Guide and Solutions Manual provides solutions, tests, exams, and teaching tips; Videotape Series provides coverage of most topics; TestGen-EQ; PowerPoint Lecture Presentation TECH: Graphing Calculator and Excel Spreadsheet Manual; Digital Video Tutor; Addison-Wesley Math Tutor Center (www.aw-bc.com/tutorcenter); MathXL; MyMathLab; InterActMath (www.interactmath.com)
—Basic properties of definite integrals. (Examples include additivity and linearity.)	SE/TE: 403–417 TR: Instructor's Resource Guide and Solutions Manual provides solutions, tests, exams, and teaching tips; Videotape Series provides coverage of most topics; TestGen-EQ; PowerPoint Lecture Presentation TECH: Graphing Calculator and Excel Spreadsheet Manual; Digital Video Tutor; Addison-Wesley Math Tutor Center (www.aw-bc.com/tutorcenter); MathXL; MyMathLab; InterActMath (www.interactmath.com)
Applications of integrals. Appropriate integrals are used in a variety of applications to model physical, biological, or economic situations. Although only a sampling of applications can be included in any specific course, students should be able to adapt their knowledge and techniques to solve other similar application problems. Whatever applications are chosen, the emphasis is on using the integral of a rate up an approximating Riemann sum and representing its limit as a definite integral. To provide a common foundation, specific applications should include finding the area of a region, the volume of a solid with known cross sections, the average value of a function, and the distance traveled by a particle along a line.
Fundamental Theorem of Calculus
—Use of the Fundamental Theorem to evaluate definite integrals	SE/TE: 417–428 TR: Instructor's Resource Guide and Solutions Manual provides solutions, tests, exams, and teaching tips; Videotape Series provides coverage of most topics; TestGen-EQ; PowerPoint Lecture Presentation TECH: Graphing Calculator and Excel Spreadsheet Manual; Digital Video Tutor; Addison-Wesley Math Tutor Center (www.aw-bc.com/tutorcenter); MathXL; MyMathLab; InterActMath (www.interactmath.com)
—Use of the Fundamental Theorem to represent a particular antiderivative, and the analytical and graphical analysis of functions so defined.	SE/TE: 420–428 TR: Instructor's Resource Guide and Solutions Manual provides solutions, tests, exams, and teaching tips; Videotape Series provides coverage of most topics; TestGen-EQ; PowerPoint Lecture Presentation TECH: Graphing Calculator and Excel Spreadsheet Manual; Digital Video Tutor; Addison-Wesley Math Tutor Center (www.aw-bc.com/tutorcenter); MathXL; MyMathLab; InterActMath (www.interactmath.com)
Techniques of antidifferentiation
—Antiderivatives following directly from derivatives of basic functions	SE/TE: 382–394 TR: Instructor's Resource Guide and Solutions Manual provides solutions, tests, exams, and teaching tips; Videotape Series provides coverage of most topics; TestGen-EQ; PowerPoint Lecture Presentation TECH: Graphing Calculator and Excel Spreadsheet Manual; Digital Video Tutor; Addison-Wesley Math Tutor Center (www.aw-bc.com/tutorcenter); MathXL; MyMathLab; InterActMath (www.interactmath.com)
—Antiderivatives by substitution of variables (including change of limits for definite integrals)	SE/TE: 394–403 TR: Instructor's Resource Guide and Solutions Manual provides solutions, tests, exams, and teaching tips; Videotape Series provides coverage of most topics; TestGen-EQ; PowerPoint Lecture Presentation TECH: Graphing Calculator and Excel Spreadsheet Manual; Digital Video Tutor; Addison-Wesley Math Tutor Center (www.aw-bc.com/tutorcenter); MathXL; MyMathLab; InterActMath (www.interactmath.com)
Applications of antidifferentiation
—Finding specific antiderivatives using initial conditions, including applications to motion along a line	SE/TE: 389–391, 393, 399–401, 402–403 TR: Instructor's Resource Guide and Solutions Manual provides solutions, tests, exams, and teaching tips; Videotape Series provides coverage of most topics; TestGen-EQ; PowerPoint Lecture Presentation TECH: Graphing Calculator and Excel Spreadsheet Manual; Digital Video Tutor; Addison-Wesley Math Tutor Center (www.aw-bc.com/tutorcenter); MathXL; MyMathLab; InterActMath (www.interactmath.com)
—Solving separable differential equations and using them in modeling. In particular, studying the equation y′ = ky and exponential growth.	SE/TE: 460–468 TR: Instructor's Resource Guide and Solutions Manual provides solutions, tests, exams, and teaching tips; Videotape Series provides coverage of most topics; TestGen-EQ; PowerPoint Lecture Presentation TECH: Graphing Calculator and Excel Spreadsheet Manual; Digital Video Tutor; Addison-Wesley Math Tutor Center (www.aw-bc.com/tutorcenter); MathXL; MyMathLab; InterActMath (www.interactmath.com)
—Numerical approximation to definite integrals. Use of Riemann and trapezoid sums to approximate definite integrals of functions represented algebraically, graphically, and by tables of values.	SE/TE: 403–417, 438–448 TR: Instructor's Resource Guide and Solutions Manual provides solutions, tests, exams, and teaching tips; Videotape Series provides coverage of most topics; TestGen-EQ; PowerPoint Lecture Presentation TECH: Graphing Calculator and Excel Spreadsheet Manual; Digital Video Tutor; Addison-Wesley Math Tutor Center (www.aw-bc.com/tutorcenter); MathXL; MyMathLab; InterActMath (www.interactmath.com)