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Student Mathematics Handbook K J. S ARL MITH Gerald L. Bradley Karl J. Smith A Student Mathematics Handbook KARL J. SMITH Cf al-∣ cS EuC O N Dl1 uED IT IsO N Gerald L. Bradley Karl J. Smith PRENTICE HALL, Upper Saddle River, NJ 07458 Executive Editor: George Lobell Special Projects Manager: Barbara A. Murray Production Editor: Shea Oakley Supplement Cover Manager: Paul Gourhan Supplement Cover Designer: Liz Nemeth Manufacturing Buyer: Alan Fischer Editorial Assistant: Gale A. Epps © 1998 by Prentice-Hall, Inc. Simon & Schuster / A Viacom Company Upper Saddle River, NJ 07458 All rights reserved. No part of this book may be reproduced, in any form or by any means, without permission in writing from the publisher Printed in the United States of America 10 9 8 7 6 5 4 3 ISBN □ -1 3 -0 a iΠ 5 4 -cl Prentice-Hall International (UK) Limited, London Prentice-Hall of Australia Pty. Limited, Sydney Prentice-Hall Canada, Inc., London Prentice-Hall Hispanoamericana, S.A., Mexico Prentice-Hall of India Private Limited, New Delhi Prentice-Hall of Japan, Inc., Tokyo Simon & Schuster Asia Pte. Ltd., Singapore Editora Prentice-Hall do Brazil, Ltda., Rio de Janeiro CONTENTS CHAPTER 1: Review of Geometry 1 1.1 Polygons 1.2 Circles 1.3 Solid Geometry 1.4 Congruent Triangles 1.5 Similar Triangles 1.6 Problem Set 1 CHAPTER 2: Review of Algebra 15 2.1 Real Numbers 2.2 Powers and Roots 2.3 Sequences and Series 2.4 Completing the Square 2.5 Solving Equations 2.6 Solving Inequalities 2.7 Determinants 2.8 Functions 2.9 Polynomials 2.10 Problem Set 2 CHAPTER 3: Review of Trigonometry 38 3.1 Trigonometric Functions 3.2 Inverse Trigonometric Functions 3.3 Evaluating Trigonometric Functions 3.4 Trigonometric Graphs 3.5 Trigonometric Identities 3.6 Solving Triangles 3.7 Trigonometric Equations 3.8 Problem Set 3 CHAPTER 4: Conic Sections 51 4.1 Standard Position Parabolas 4.2 Translation of Parabolas 4.3 Representation of Parabolas in Polar Form 4.4 Parabolic Reflectors 4.5 Problem Set 4 CHAPTER 5: Conic Sections: The Ellipse and the Hyperbola 63 5.1 Ellipses 5.2 Hyperbolas 5.3 Eccentricity and Polar Coordinates 5.4 Problem Set 5 CHAPTER 6: Curve Sketching 80 6.1 Symmetry 6.2 Extent 6.3 Asymptotes 6.4 Intercepts 6.5 Problem Set 6 CHAPTER 7: Catalog of Special Curves 94 CHAPTER 8: Limit Formulas 104 8.1 Definition of Limit 8.2 Rules of Limits 8.3 Limits of a Function of Two Variables CHAPTER 9: Differentiation Formulas 109 9.1 Definition of Derivative 9.2 Procedural Rules of Differentiation 9.3 Differentiation Rules 9.4 Functions of Two Variables CHAPTER 10: Integration Formulas 113 10.1 Definition of Integral 10.2 Procedural Rules for Integration 10.3 Integration Rules 10.4 Table of Integrals 10.5 Bernoulli and Euler Numbers; Gamma and Beta Functions 10.6 Definite Integral Formulas CHAPTER 11: Series 176 11.1 Series of Constants 11.2 Taylor Series APPENDICES: 181 A. Mathematical Symbols B. Greek Alphabet C. Answers INDEX: 195 PREFACE Calculus is probably the first mathematics course you have taken that is not self-contained in the sense that the material from previous mathematics courses is expected without specifically mentioning it in the textbook. This supplement is a convenient reference book to be used along with your textbook, Calculus, to remind you of those formulas or topics that you may have forgotten. This handbook is organized so that it can be used in two ways. The first use is as a reference manual providing a summary of terminology, formulas, and tables, not only of prerequisite mathematics, but also of the material covered in a standard calculus course. The second use is as a brief review of material assumed as a prerequisite for a course in calculus. This material is presented with examples, brief written exposition, and practice problems. The topics that are included with exposition and practice problems are sometimes missing from the backgrounds of many students who otherwise have the prerequisites for calculus. We remind the student that nearly everyone qualified to enroll in calculus has, for a variety of reasons, gaps in knowledge of prerequisite material, and the brief review in this handbook can help to bridge that gap. We provide this supplement to Calculus free of charge with the purchase of the textbook, in an attempt to help ensure your success in calculus. Use this book for reference, and as a handbook as you progress through the course. In addition, new technology has changed the emphasis of many of the topics in a calculus course. One recent change is the acknowledgement of calculators and computers to help not only with the mechanics of algebra, but also with the mechanics of differentiation and integration in calculus. Outside of the academic environment, engineers and physicists tells us that using available technology, as well as tables of integration, is by far more important than many of the esoteric topics they were taught in their calculus courses many years ago. For that reason, calculus books are evolving, and the emphasis is not on obscure esoteric topics, but rather practical knowledge that balances between application and theory. To capture this new emphasis, we see the need for you to have a complete integration table, so one has been provided in this handbook. Mathematics Handbook for CALCULUS Page 1 CHAPTER 1 Review of Geometry In this book we use the following variables when stating formulas: A = area, P = perimeter, C = circumference, S = surface area, and V = volume. Also, r denotes radius, Λ altitude, I slant height, b base, B area of base, θ central angle expressed in radians. 1.1_____ Polygons CLASSIFICATION TRIANGLES Type Number of sides A = ⅛bh The sum of the measures of the triangle 3 angles of a triangle is 180°. quadrilateral 4 pentagon 5 hexagon 6 heptagon 7 octagon 8 nonagon 9 decagon 10 undecagon 11 dodecagon 12 Pythagorean theorem If angle C is a right angle, then c2 = α2 + ⅛2. 45°-45°-90° triangle theorem For any right triangle with acute angles measuring 45°, the legs are the same length, and the hypotenuse has a length equal to ∖∕2 times the length of one of those legs. 30°-60°-90° triangle theorem For any right triangle with acute angles measuring 30° and 60°: 1. The hypotenuse is twice as long as the leg opposite the 30° angle (the shorter leg). 2. The leg opposite the 30° angle (the shorter leg) is ∣ as long as the hypotenuse. 3. The leg opposite the 60° angle (the longer leg) equals the length of the other (shorter) leg times y∕⅛ 4. The leg opposite the 30° angle equals the length of the other leg divided by √37 Equilateral triangle For any equilateral triangle: a = β = γ = 60° A = ⅜δ2√3 h = ∣6√3 Page 2 Chapter 1 QUADRILATERALS Rectangle Square Parallelogram Trapezoid A = (w A = s2 A = bh — α⅛sin θ A — ⅛h(a + 6) P = 2( + 2w P = 4,s P = 2a + 2b P = a + b REGULAR POLYGON OF n SIDES A = ∣nα2cot A P — an 1.2 Circles TERMINOLOGY Definition: In a plane, a circle is the set of all points a given distance, called the radius, from a given point, called the center. Circumference: distance around a circle. Chord: a line joining two points of a circle. Diameter: a chord through the center: AB is Figure 1.1. Arc: part of a circle: BC, AC, or ACB in Figure 1.1. The length s of an arc of a circle of radius r with central angle θ (measured in radians) is .s = rθ. To intercept an arc is to cut off the arc; in Figure 1.1, ACOB intercepts BC. Tangent of a circle is a line that intersects the circle at one and only one point. Figure 1.1 Secant of a circle is a line that intersects the circle at exactly two points. Inscribed polygon is a polygon all of whose sides are chords of a circle. A regular inscribed polygon is a polygon all of whose sides are the same length. Inscribed circle is a circle to which all the sides of a polygon are tangents. Circumscribed polygon is a polygon all of whose sides are tangents to a circle. Circumscribed circle is a circle passing through each vertex of a polygon. Mathematics Handbook for CALCULUS Page 3 BASIC FORMULAS Circle Sector Segment A = ιrr2 X - 1∕⅜ A = ∣r2(0 — sin 0) C = 2τrr — πd 1.3___________ Solid Geometry Rectangular parallelepiped (box) Prism Pyramid V = abc V= Bh V=±Bh Diagonal = ∖∕α2 + ⅛2 + c2. B is area of the base B is area of the base Hexagonal prism Tetrahedron Right circular cylinder Right circular cone (a pyramid with a V = πr2h V — ^τrr2h triangular base) Lateral surface = 2πrh Lateral surface = πrl V = ∣⅛∙γ∕ s(s — α)(s — ⅛)(s — c) S = 2πrħ + 2τrr2 S = πrl + πr2 where s = ∣(a + b + c) Page 4 Chapter 1 Frustum of a right circular cone Frustum of a pyramid V = ∣7rA(r2 + rR + ∕i2) or y=∣λ(5 √⅞¾ + B ) 1 + 2 V = ∣A(B + ∖∕B B ÷ B ) 1 1 2 2 Torus Spherical segment S = 4π2Rr S — 2πrh, with radius r V= 2τr2Rr2 V = ∣τrA(3r 2 + 3r 2 + ⅛2) 1 2 For the “cap” S = 2πrh F=fΛ(3r 2 ÷ Λ) 2 Cylinder with a cross-sectional area A Prismatoid, pontoon, wedge V = Ah; S=pt + 2A V = ∣A(5 + 4B + B ) 0 1 2