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10.1 De nition and Properties 10.2 One-Sided Continuity 10.3 Continuity over a Closed Interval
82 83 84
11 12 13
THE SLOPE OF A TANGENT LINE THE DERIVATIVE MORE ON THE DERIVATIVE
13.1 Differentiability and Continuity 13.2 Further Rules for Derivatives
91 97 105
105 105
14
MAXIMUM AND MINIMUM PROBLEMS
14.1 Relative Extrema 14.2 Absolute Extrema
110 111
15
THE CHAIN RULE
15.1 Composite Functions 15.2 Differentiation of Composite Functions
122 123
16 17
IMPLICIT DIFFERENTIATION THE MEAN-VALUE THEOREM AND THE SIGN OF THE DERIVATIVE
17.1 Rolle s Theorem and the Mean-Value Theorem 17.2 The Sign of the Derivative
137 138
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CONTENTS
1
COORDINATE SYSTEMS ON A LINE
1.1 1.2 The Coordinates of a Point Absolute Value
2
COORDINATE SYSTEMS IN A PLANE
2.1 2.2 2.3 The Coordinates of a Point The Distance Formula The Midpoint Formulas
8 10 11
3 4
GRAPHS OF EQUATIONS STRAIGHT LINES
4.1 4.2 4.3 4.4 Slope Equations of a Line Parallel Lines Perpendicular Lines
15 25
25 27 30 31
5 6
INTERSECTIONS OF GRAPHS SYMMETRY
6.1 6.2 Symmetry about a Line Symmetry about a Point
38 44
44 45
7
FUNCTIONS AND THEIR GRAPHS
7.1 7.2 7.3 7.4 The Notion of a Function Intervals Even and Odd Functions Algebra Review: Zeros of Polynomials
49 52 53 55
CONTENTS
18 19 20 21
RECTILINEAR MOTION AND INSTANTANEOUS VELOCITY INSTANTANEOUS RATE OF CHANGE RELATED RATES
145 152 156
APPROXIMATION BY DIFFERENTIALS; NEWTON S METHOD 164
21.1 Estimating the Value of a Function 21.2 The Differential 21.3 Newton s Method 164 165 166
22 23
HIGHER-ORDER DERIVATIVES APPLICATIONS OF THE SECOND DERIVATIVE AND GRAPH SKETCHING
23.1 Concavity 23.2 Test for Relative Extrema 23.3 Graph Sketching
178 180 182
24 25
MORE MAXIMUM AND MINIMUM PROBLEMS ANGLE MEASURE
25.1 Arc Length and Radian Measure 25.2 Directed Angles
190 197
197 199
26
SINE AND COSINE FUNCTIONS
26.1 General De nition 26.2 Properties
203 205
27
GRAPHS AND DERIVATIVES OF SINE AND COSINE FUNCTIONS
27.1 Graphs 27.2 Derivatives
215 218
28
THE TANGENT AND OTHER TRIGONOMETRIC FUNCTIONS
CONTENTS
29
ANTIDERIVATIVES
29.1 De nition and Notation 29.2 Rules for Antiderivatives
235 236
30
THE DEFINITE INTEGRAL
30.1 Sigma Notation 30.2 Area under a Curve 30.3 Properties of the De nite Integral
243 243 246
31
THE FUNDAMENTAL THEOREM OF CALCULUS
31.1 Calculation of the De nite Integral 31.2 Average Value of a Function 31.3 Change of Variable in a De nite Integral
253 254 255
32
APPLICATIONS OF INTEGRATION I: AREA AND ARC LENGTH
32.1 Area between a Curve and the y-Axis 32.2 Area between Two Curves 32.3 Arc Length
264 265 267
33
APPLICATIONS OF INTEGRATION II: VOLUME
33.1 Solids of Revolution 33.2 Volume Based on Cross Sections
273 276
34
THE NATURAL LOGARITHM
34.1 De nition 34.2 Properties
285 286
35
EXPONENTIAL FUNCTIONS
35.1 Introduction 35.2 Properties of ax 35.3 The Function ex
292 292 293
36
L H PITAL S RULE; EXPONENTIAL GROWTH AND DECAY
36.1 L H pital s Rule 36.2 Exponential Growth and Decay
301 303
CONTENTS
37
INVERSE TRIGONOMETRIC FUNCTIONS
37.1 One-One Functions 37.2 Inverses of Restricted Trigonometric Functions
309 311
38 39
INTEGRATION BY PARTS TRIGONOMETRIC INTEGRANDS AND TRIGONOMETRIC SUBSTITUTIONS
39.1 Integration of Trigonometric Functions 39.2 Trigonometric Substitutions
329 331
40 Appendix A Appendix B Appendix C Appendix D Appendix E Appendix F
INTEGRATION OF RATIONAL FUNCTIONS; THE METHOD OF PARTIAL FRACTIONS TRIGONOMETRIC FORMULAS BASIC INTEGRATION FORMULAS GEOMETRIC FORMULAS TRIGONOMETRIC FUNCTIONS NATURAL LOGARITHMS EXPONENTIAL FUNCTIONS
339 347 348 349 350 352 354 356 393
ANSWERS TO SUPPLEMENTARY PROBLEMS INDEX
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Coordinate Systems on a Line
1.1 THE COORDINATES OF A POINT Let L be a line. Choose a point O on the line and call this point the origin.
Now select a direction along L ; say, the direction from left to right on the diagram.
For every point P to the right of the origin O, let the coordinate of P be the distance between O and P.
(Of course, to specify such a distance, it is rst necessary to establish a unit distance by arbitrarily picking two points and assigning the number 1 to the distance between these two points.) In the diagram
the distance OA is assumed to be 1, so that the coordinate of A is 1. The point B is two units away from O; therefore, B has coordinate 2. Every positive real number r is the coordinate of a unique point on L to the right of the origin O; namely, of that point to the right of O whose distance from O is r. To every point Q on L to the left of the origin O,
Copyright 2008, 1997, 1985 by The McGraw-Hill Companies, Inc. Click here for terms of use.
COORDINATE SYSTEMS ON A LINE
[CHAP. 1
we assign a negative real number as its coordinate, the number QO, the negative of the distance between Q and O. For example, in the diagram
the point U is assumed to be a distance of one unit from the origin O; therefore, the coordinate of U is 1. The point W has coordinate 1 , which means that the distance WO is 1 . Clearly, every negative real number is the coordinate of a 2 2 unique point on L to the left of the origin. The origin O is assigned the number 0 as its coordinate. This assignment of real numbers to the points on the line L is called a coordinate system on L .
Choosing a different origin, a different direction along the line, or a different unit distance would result in a different coordinate system. 1.2 ABSOLUTE VALUE For any real number b de ne the absolute value |b| to be the magnitude of b; that is, |b| = b b if b 0 if b < 0
In other words, if b is a positive number or zero, its absolute value |b| is b itself. But if b is negative, its absolute value |b| is the corresponding positive number b. EXAMPLES
|3| = 3 5 5 = 2 2 |0| = 0 | 2| = 2 1 1 = 3 3
Properties of the Absolute Value Notice that any number r and its negative r have the same absolute value, |r| = | r| An important special case of (1.1) results from choosing r = u v and recalling that (u v) = v u, |u v| = |v u| If |a| = |b|, then either a and b are the same number or a and b are negatives of each other, |a| = |b| Moreover, since |a| is either a or a, and ( a)2 = a2 , |a|2 = a2 Replacing a in (1.4) by ab yields |ab|2 = (ab)2 = a2 b2 = |a|2 |b|2 = (|a| |b|)2 whence, the absolute value being nonnegative, |ab| = |a| |b| (1.5) (1.4) implies a = b (1.3) (1.2) (1.1)
CHAP. 1]
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