Chapter of Study
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Overview of Chapter
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Assignments, Summative and Formative Assessments
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1.1 Four Ways to Represent a Function 10
1.2 Mathematical Models: A Catalog of Essential Functions 23 1.3 New Functions from Old Functions 36 1.4 Graphing Calculators and Computers 44 1.5 Exponential Functions 51 1.6 Inverse Functions and Logarithms 58 Review 72 |
The fundamental objects that we deal with in calculus are functions. This chapter prepares the way for calculus by discussing the basic ideas concerning functions, their graphs, and ways of transforming and combining them. We stress that a function can be represented in different ways: by an equation, in a table,by a graph, or in words. We look at the main types of functions that occur in calculus and describe the
process of using these functions as mathematical models of real-world phenomena. We also discuss the use of graphing calculators and graphing software for computers. |
1.1: 2,3,7,9,15,23,25,27,29,31,33,39,41,43,47,49,73,77
1.2: 1, 3, 4, 5, 8, 11, 13, 15, 17, 19, 21, 27 Notes: APCalc 1.2 Stewart Math Models.pdf 1.3: 2, 3, 5, 9, 11, 13, 15, 17, 19, 21, 29, 31, 35, 37, 39, 41, 50, 51, 53, 55 1.4 Text: HW 1.4: 3,5,9,15,17,19,21,23,25,31 Group Work Handouts 1 and 2 1.5: 1,3,6,9,11,13,19,23,28,29,32 1.6: 1-14, 15-37 odd, 39-42, 49-54, 63-71 odd |
2.1 The Tangent and Velocity Problems 82 2.2 The Limit of a Function 87 2.3 Calculating Limits Using the Limit Laws 99 2.4 The Precise Definition of a Limit 108 2.5 Continuity 118 2.6 Limits at Infinity; Horizontal Asymptotes 130 2.7 Derivatives and Rates of Change 143 Writing Project N Early Methods for Finding Tangents 153 2.8 The Derivative as a Function 154 Review 165 3.1 Derivatives of Polynomials and Exponential Functions 174 Applied Project N Building a Better Roller Coaster 184 3.2 The Product and Quotient Rules 184 3.3 Derivatives of Trigonometric Functions 191 3.4 The Chain Rule 198 Applied Project N Where Should a Pilot Start Descent? 208 3.5 Implicit Differentiation 209 Laboratory Project N Families of Implicit Curves 217 3.6 Derivatives of Logarithmic Functions 218 3.7 Rates of Change in the Natural and Social Sciences 224 3.8 Exponential Growth and Decay 237 3.9 Related Rates 244 3.10 Linear Approximations and Differentials 250 Laboratory Project N Taylor Polynomials 256 3.11 Hyperbolic Functions 257 Review 264 4.1 Maximum and Minimum Values 274 Applied Project N The Calculus of Rainbows 282 4.2 The Mean Value Theorem 284 4.3 How Derivatives Affect the Shape of a Graph 290 4.4 Indeterminate Forms and l’Hospital’s Rule 301 Writing Project N The Origins of l’Hospital’s Rule 310 4.5 Summary of Curve Sketching 310 4.6 Graphing with Calculus and Calculators 318 4.7 Optimization Problems 325 Applied Project N The Shape of a Can 337 4.8 Newton’s Method 338 4.9 Antiderivatives 344 Review 351 5.1 Areas and Distances 360 5.2 The Definite Integral 371 Discovery Project N Area Functions 385 5.3 The Fundamental Theorem of Calculus 386 Distance and Area Lab 5.4 Indefinite Integrals and the Net Change Theorem 397 Writing Project N Newton, Leibniz, and the Invention of Calculus 406 5.5 The Substitution Rule 407 Review 415 6.1 Areas Between Curves 422 Applied Project N The Gini Index 429 6.2 Volumes 430 6.3 Volumes by Cylindrical Shells 441 6.4 Work 446 6.5 Average Value of a Function 451 Applied Project N Calculus and Baseball 455 Applied Project N Where to Sit at the Movies 456 Review 457 7.1 Integration by Parts 464 7.2 Trigonometric Integrals 471 7.3 Trigonometric Substitution 478 7.4 Integration of Rational Functions by Partial Fractions 484 7.5 Strategy for Integration 494 7.6 Integration Using Tables and Computer Algebra Systems 500 8.1 Arc Length 538 Discovery Project N Arc Length Contest 545 8.2 Area of a Surface of Revolution 545 Discovery Project N Rotating on a Slant 551 8.3 Applications to Physics and Engineering 552 Discovery Project N Complementary Coffee Cups 562 8.4 Applications to Economics and Biology 563 8.5 Probability 568 |
C2: In A Preview of Calculus we saw how the idea of a limit underlies the various branches of calculus. It is therefore appropriate to begin our study of calculus by investigating limits and their properties. The special type of limit that is used to find tangents and velocities gives rise to the central idea in differential calculus, the derivative. Challenging content, some of the limit material was seen in pre-calculus but the important "bits" underlie many of the basics of differential and integral calculus We have seen how to interpret derivatives as slopes and rates of change. We have seen how to estimate derivatives of functions given by tables of values. We have learned how to graph derivatives of functions that are defined graphically. We have used the definition of a derivative to calculate the derivatives of functions defined by formulas. But it would be tedious if we always had to use the definition, so in this chapter we develop rules for finding derivatives without having to use the definition directly. These differentiation rules enable us to calculate with relative ease the derivatives of polynomials, rational functions, algebraic functions, exponential and logarithmic functions, and trigonometric and inverse trigonometric functions. We then use these rules to solve problems involving rates of change and the approximation of functions. |
5.1 Areas and Distance Text 1, 3, 5, 7,10, 15, 17, 5.2 The Definite Integral Text 1, 5, 7, 9, 17, 23, 29, 33, 35, 37, 47, 55, 59 5.2 Video Notes Link 5.3 The Fundamental Theorem of Calculus Text 3, 5, 7-45 odd, 55-59 odd Video Tutorial 5.3 (Links to an external site.) 5.4 Text Problems 1-71 every other odd(video) and Field Trip Lab TLT Tube Video Notes 5.4 5.5 Text 3, 5, 11, 13, 17, 21, 23, 33, 39, 45, 47, 53, 59 |