u with Trigonometry and Analytic Geometry

John Saxon with

Diana Harvey

Frank Wang

SAXON PUBLISHERS, INC. Used by Permission

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Calculus with Trigonometry and Analytic Geometry

Copyright© 1988 by Saxon Publishers, Inc.

No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher.

Printed in the United States of America

ISBN: 0-939798-34-4

Editor and production supervisor: Nancy Warren Computer-generated line art: Kennebec Laser Compositor: Black Dot Graphics

Sixth printing: May 1995

Printed on recycled paper

Saxon Publishers, Inc. 1320 W. Lindsey Norman, Oklahoma 73069

Used by Permission

Sample

Lesson A

Lesson B

Lesson 1

Contents

Preface

The real numbers · Fundamental concept review More concept review · Geometry review Deductive reasoning · The contrapositive · Converse and mverse

Lesson 2 Radian measure of angles · Trigonometric ratios · Four quadrant signs · Simplifying trigonometric expressions

Lesson 3 Word problem review Lesson 4

Lesson 5

Lesson 6 Lesson 7

Lesson 8

Lesson 9 Lesson 10

Lesson 11

Lesson 12

Lesson 13 Lesson 14

Lesson 15

Lesson 16 Lesson 17

Lesson 18

Lesson 19 Lesson 20 Lesson 21 Lesson 22

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Lesson 25

Functions: Their equations and graphs · Functional notation · Domain and range The unit circle · Graphing sinusoids Similar triangles · Functions of - 0

Quadratic equations Pythagorean identities · Trigonometric identities · Cofunctions Abstract word problems Important numbers · Exponential functions Polar coordinates (vectors) · Polar coordinates (complex numbers) Absolute value as a distance · The line as a locus · The circle as a locus Special functions The logarithmic form of the exponential · Base 10 and base e · Simple logarithm problems Evaluating polynomials Continuity · Left-hand and right-hand limits Sum and difference identities for trigonometric functions · Double-angle identities for sine and cosine Graphs of logarithmic functions · Period of a function Limit of a function The parabola as a locus · Translated parabolas Inverse trigonometric functions · Trigonometric equations Interval notation · Products of linear factors · Tangents · Increasing and decreasing functions Logarithms of products and quotients · Logarithms of powers · Exponential equations Infinity as a limit · Undefined limits Sums, products, and quotients of functions · Composition of functions

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Locus development · Equation of the ellipse · Foci The derivative Change of base · Logarithmic inequalities Translation of functions · Rational functions I The hyperbola Binomial expansion · Recognizing the equations of conic sections Roots of complex numbers · Trigonometric functions. of n0

The derivative of xn · Notations for the derivative Identities for the tangent function · Area and volume

The constant-multiple rule · The derivatives of sums and differences Exponential growth and decay Derivative of ex and In lxl · Derivative of sin x and cos x Equation of the tangent line · Higher-order derivatives Graphs of rational functions II · A special limit Newton and Leibniz · The differential Graph of tan 0 · Graphs of reciprocal functions Product rule for derivatives and differentials · Proof of the product rule An antiderivative · Integration Factors of polynomial functions · Graphs of polynomial functions Implicit differentiation The integral of a constant · Integral of Cf(x) · Integral of xn Critical numbers Differentiation by u substitution Integral of a sum · Integral of l

X

Units for the derivative · Normal lines Graphs of rational functions III · Repeated factors The derivative of a quotient · Proof of the quotient rule Area under a curve The chain rule · Equivalent forms for the derivative Using f' to characterize f · Using f' to define max and min Related rate problems Fundamental theorem of integral calculus Derivatives of trigonometric functions · Summary of rules for derivatives and differentials Concavity and inflection points · Applications of the second derivative

Contents

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199 205

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246 250 254

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Derivatives of composite functions · Derivatives of products and quotients Integration by guessing Maximization and minimization problems Riemann sum · The definite integral Velocity and acceleration (motion I) · Motion due to gravity More integration by guessing Properties of the definite integral Explicit and implicit equations · Inverse functions Computing areas Area between two curves Game playing with f, f', and f"

Applications of the definite integral I Critical number ( closed interval) theorem Derivatives of inverse trigonometric functions · What to memorize Falling body problems U substitution · Change of variable · Proof of the substitution theorem Functions of y

Even and odd functions Integration by parts Properties of limits · Some special limits

Solids of revolution Derivatives and integrals of ax and loga x · Derivative of lxl Fluid force Continuity of functions Integration of odd powers of sin x and cos x Applications of the definite integral (work II) Particle motion III L'Hopital's rule · Proof of L'Hopital's rule Asymptotes of rational functions Balance points Volume by washers Limits and continuity · Differentiability Integration of even powers of sin x and cos x Centroids Logarithmic differentiation The mean value theorem · Application of the mean value theorem · Proof of Rolle's theorem

Contents

301

305 309 314 318 322 326 332 340 343 347 352

356 361

367 373

378 381 385 389

394 397 403 408 414 418

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427 431 435 440

444 450 453 457 462

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Appendix

viii

Rules for even and odd functions Volume by shells Separable differential equations Average value of a function · Mean value theorem for integrals Particle motion IV Derivatives of inverse functions Solids of revolution IV Absolute value Integral of tann x · Integral of cotn x

Second fundamental theorem of integral calculus · The natural logarithm function Approximation with differentials Limit of sin x • A note ( optional)

X

Integrals of sec u and csc u · Trig substitution Polar equations · Polar graphing Partial fractions I Polar graphing II Partial fractions II Integration by parts II Implicit differentiation II Partial fractions III Derivative of ex and of In x · Derivative of sin x Proofs of the fundamental theorem · Epsilon delta proofs Answers to odd-numbered problems

Important formulas, facts, and rules Index

Contents

469 474

477 483 489

493 498 504

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519 522 528

533 541

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550 553

557 562

565 571

580 613 619

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LESSON J

1.A Deductive reasoning

14

Deductive reasoning · The contrapositive · Converse and inverse

lesson 1

Inductive reasoning is the process of determining or formulating a general rule by extrapolating from a number of individual observations or by just guessing. This most imprecise process produces rules that may or may not be true. A child who goes fishing in a pond and catches only fish that have scales might induce that all the fish in that pond have scales. This would be a guess at best because the child has not seen every fish in the pond and some might be catfish, a fish that does not have scales. Deductive reasoning is quite different. Deductive reasoning is the process of applying a given rule logically so that a mistake is not made in the application of the rule. Deductive reasoning does not consider the truth or falsity of the rule being applied.

The ancient Greeks formalized the study oflogic with their use of syllogisms in their investigation of deductive reasoning. A syllogism is a formal reasoning process in which a conclusion is inferred from two statements called premises. We will look at syllogisms in which the premises are called categorical propositions because they place things in categories. We will concentrate on premises called universal affirmatives. These premises affirm that all members of a certain set possess a certain property. Such a premise is often called the major premise. The other premise is often called the minor premise and identifies a member of this set. The conclusion follows that this member has the property possessed by all the members of the set. We demonstrate by using one of the oldest syllogisms known.

All men are mortal. (Major premise)

This statement establishes mortality as a property possessed by every member of the set of all men.

Aristotle is a man. (Minor premise)

This statement identifies Aristotle as a member of the set of all men.

Aristotle is mortal. (Conclusion)

This conclusion is a logical consequence because if Aristotle is a member of the set of all men, then he possesses the properties possessed by every member of this set.

We use this type of reasoning in geometric proofs. Observe:

The sum of the exterior angles of a convex polygon is 360°.

Triangle ABC is a convex polygon.

The sum of the exterior angles of triangle ABC is 360°.

The major premise identifies a property of every member of the set of convex polygons. The minor premise identifies triangle ABC as a member of the set of convex polygons. Thus, triangle ABC has all the properties possessed by every member of the set of convex polygons. The entire three-step process is called an argument. In our investigation of syllogistic reasoning we will concentrate on the argument and will not consider the truth or falsity of the major premise. Consider the following syllogism:

i , 'l

i

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15

All frogs are green.

Henry is a frog.

Henry is green.

(Major premise)

(Minor premise)

(Conclusion)

1.A Deductive reasoning

The argument is a valid argument because the major premise states a property possessed by all frogs (they are green), and the minor premise identifies Henry as a member of the set of all frogs. The major premise is false because some frogs are not green and thus Henry might be brown or red or some other color. However, because the argument is a valid argument, we will say that the conclusion is a valid conclusion. This does not mean that the conclusion is true. It means only that the argument is valid.

The conclusion in the following syllogism is invalid because the argument is faulty, as the minor premise does not identify a member of the set defined by the major premise.

If it rains, I will go to town.

It did not rain.

I did not go to town.

(Major premise)

(Minor premise)

(Conclusion)

The major premise identifies an action I will take on each member of the set of days on which rain occurs. It makes no statement about what I will do on days on which there is no rain. Thus, any conclusion about what happens on dry days would be invalid. The day in question is not a member of the set of rainy days and may or may not possess the property in question.

Example 1.1 Is the following argument a valid argument?

All normal dogs have four legs.

That dog has four legs.

That dog is a normal dog.

Solution The argument is invalid. The set described is the set of normal dogs. For a valid argument, the minor premise should have stated that a particular dog was a normal dog.

Example 1.2 Is the following argument valid?

All boys are good.

That child is a good child.

That child is a boy.

Solution Invalid. The major premise makes a statement about the set of all boys. The minor premise talks about a member of the set of good children. To be a valid argument, the minor premise would have to identify a member of the set of all boys.

Example 1.3 Is the following argument a valid argument?

All chickens have three legs.

Henny Penny is a chicken.

Henny Penny has three legs. Used by Permission

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16 Lesson 1

Solution Valid. All chickens do not have three legs, but we still say that the argument is a valid argument. This does not mean that the conclusion is true. It means that the argument is valid because the major premise identifies a property of the set of all chickens and the minor premise identifies Henny Penny as a member of the set of all chickens.

1.B The

contrapositive The major premise either uses or implies an if-then statement that has two parts called the hypothesis and the conclusion. The hypothesis begins with the word if and the conclusion begins with the word then. When the words zf and then are not written, the premise can be rewritten so that these words are used. For example, the

· major premise "Rabbits are fast runners" can be written as an if-then statement by writing: If an animal is a rabbit, then the animal is a fast runner.

HYPOTHESIS CONCLUSION

If an animal is a rabbit, then the animal is a fast runner

We can make exactly the same statement another way by turning the statement around and using negatives. We call this alternate statement of the same premise the contrapositive. Two steps are necessary to form the contrapositive of a premise. The first step is to replace the if statement with the negative of the then statement. The second step is to replace the then statement with the negative of the zf statement. Thus, the contrapositive of this premise is:

If an animal is not a fast runner, then the animal is not a rabbit.

If a premise is true, its contrapositive is also true. If a premise is false, its contrapositive is also false. This is easy to see if we use a Venn diagram for this example.

F F

We place all rabbits (R) inside the smaller curve and all fast animals (F) inside the larger curve. This puts all animals who are not fast (F) outside the larger curve. From this we can see that if an animal is a rabbit, it is also inside the F curve and is a fast runner. Also, we see that if an animal is not a fast runner, it is outside the larger curve and thus cannot be a rabbit.

Example 1.4 Is the following argument valid?

All nonathletes are vegetarians.

Jim is a nonvegetarian.

Jim is an athlete.

(Major premise)

(Minor premise)

(Conclusion)

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17 1.C Converse and inverse

Solution If we write the contrapositive of the major premise, we get:

1.C

All nonvegetarians are athletes.

Jim is a nonvegetarian.

Jim is an athlete.

(Major premise)

(Minor premise)

(Conclusion)

This is a valid argument, so the original argument is also valid. The conclusion is not necessarily true, but the argument is valid.

Converse We will call the initial statement of a premise the conditional. If we turn the and inverse conditional around and use negatives, we form the contrapositive, which allows us to

make the conditional statement another way. If we turn the conditional statement around and do not use negatives, we do not make the same statement. This new statement is called the converse. Ifwe wish, we can turn the converse around and use negatives to make the converse statement another way. We call this form the inverse.

The confusion that results from these four forms has delighted logicians since the time of the ancient Greeks. There are two ways in which the statements in the hypothesis and the conclusion can be written, and each of these statements can be made in an alternative form. We note that the truth or falsity of the converse does not depend on the truth or falsity of the conditional.

Conditional: If an animal is a rabbit, then the animal is a fast runner. True.

Converse: If an animal is a fast runner, then the animal is a rabbit. False.

The conditional is true, but the converse is false. If the conditional is true, its partner, the contrapositive, will be true. If the converse is false, its partner, the inverse, will be false. It is helpful to think of these equivalent statements in pairs, as we show in the accompanying table.

if then if-then

Conditional rabbit fast p ~ Q. Contrapositive not fast not rabbit Q ~ p

Converse fast rabbit Q ~ p

Inverse not rabbit not fast p ~ Q

On the right we use P and Q to represent the original hypothesis and conclusion. We indicate the negation of P, or "not P," by writing P, and we write "not Q" as Q. The paired statements in the above table use different words to say the same thing. If one statement is true, the other is true. If one statement is false, the other is false. We note that the converse is a simple reversal of P and Q and that the inverse is the converse stated using negations.

The converse and inverse are important in mathematics because sometimes they are true and sometimes they are not true. For example, consider the statements about polygons listed in the following table.

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18 Lesson 1

if then zfthen

Conditional a square a quadrilateral p ------,> Q_ true Contrapositive not a quadrilateral not a square Q ------,> p true

Converse a quadrilateral a square Q_ ------,> p false Inverse not a square not a quadrilateral p ------,> Q false

In the example above the converse and inverse are false. Now consider these statements about the lengths of sides and the measures of

the angles opposite these sides in a triangle.

if then if-then

Conditional Sides have equal lengths Angles have equal measures p ------,> Q_ true Contrapositive Angles do not have Sides do not have Q ------,> p true

Converse Inverse

equal measures equal lengths

Angles have equal measures Sides have equal length Q_ ------,> p true Sides do not have Angles do not have p ------,> Q true

equal lengths equal measures

Mathematical statements in which all four forms are true are called if and only if statements, and the abbreviation ifI is often used to mean if and only if. When we use the words if and only if, we are making two statements that symbolically have arrowheads going in both directions.

iff means both P ~ Q and P ~ Q

Problem set 1 Are following arguments valid or invalid?

1. All gurus are Asian. 2. All gurus are Asian. Frank is a guru. Alfred is Asian.

Therefore, Frank is Asian. Therefore, Alfred is a guru.

3. State the contrapositive of the following conditional statement: If the light is on, then the switch is on.

4. Johnny was told that pleads to q. He walked down the hall and saw "not q." Did the presence of "not q" imply p or "not p" or imply nothing at all? Explain.

5. Find the distance from the point (6, 2) to the midpoint of the line segment whose endpoints are (7, 2) and ( - 3, 8).

6. Write the slope-intercept form of the equation of the line which passes through the point (2, 2) and which is perpendicular to the line 2y - x - 1 = 0.

7. Write the double-intercept form of the equation of the line which passes through the points ( 1, 3) and (-1, 7).

8. Complete the square to rewrite x 2 = -6x - 13 in the form (x + a) 2 + b = 0, where a and b are constants.

9. Use the quadratic formula to find all the values of x which make x 2 - 3x + 7 equal to zero.

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19

{2y2 - x 2 = 1

10. Solve: y + 1 = x

Problem set 1

11. Divide x 3 - 13x2 + lOx - 8 by X - 1.

m+b l(a b) 12. Solve for R 1: -c- = k Ri + R2

Simplify:

4 - 2Y3 13. ~ r;;,

2 - V 3

15. s/t-2~+\/84

17. 1

1 1 +----

1 +-1-

Factor:

21. Simplify:

1 + l 2

a2 - b2 a+b

41! 23. :Evaluate:

38131

CONCEPT REVIEW 24. If x > y, then compare: A. l X

3i2 + 2i + 4 14· 3i - 7 + 2i

16. ~ yll4X3!2

18. m p

x+---1 - L

m

20. 2x3 + 3x2 - 2x 4

22. Evaluate: L (j2 - 2j) j-1

B. _!_ y

Find the value of x in each of the following figures:

25. --------- 26.

27. Solve for x and y.

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