Complete Solutions Manual for Calculus of a Single Variable, Volume 1. Calculus ELEVENTH EDITION

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1 Complete Solutions Manual for Calculus of a Single Variable, Volume Calculus ELEVENTH EDITION Cengage Learning. All rights reserved. No distribution allowed without epress authorization. Ron Larson The Pennslvania Universit, The Behrend College

2 0 Cengage Learning ALL RIGHTS RESERVED. No part of this work covered b the copright herein ma be reproduced, transmitted, stored, or used in an form or b an means graphic, electronic, or mechanical, including but not limited to photocoping, recording, scanning, digitizing, taping, Web distribution, information networks, or information storage and retrieval sstems, ecept as permitted under Section 07 or 0 of the 97 United States Copright Act, without the prior written permission of the publisher. For product information and technolog assistance, contact us at Cengage Learning Customer & Sales Support, For permission to use material from this tet or product, submit all requests online at Further permissions questions can be ed to permissionrequest@cengage.com. ISBN-: ISBN-0: Cengage Learning 0 Channel Center Street Boston, MA 00 USA Cengage Learning is a leading provider of customized learning solutions with office locations around the globe, including Singapore, the United Kingdom, Australia, Meico, Brazil, and Japan. Locate our local office at: Cengage Learning products are represented in Canada b Nelson Education, Ltd. To learn more about Cengage Learning Solutions, visit Purchase an of our products at our local college store or at our preferred online store Printed in the United States of America Print Number: 0 Print Year: 07

3 Contents Chapter P: Preparation for Calculus... Chapter : Limits and Their Properties... Chapter : Differentiation... Chapter : Applications of Differentiation... Chapter : Integration... Chapter : Logarithmic, Eponential, and Other Transcendental Functions... Chapter : Differential Equations... 7

4 CHAPTER P Preparation for Calculus Section P. Graphs and Models... Section P. Linear Models and Rates of Change... 0 Section P. Functions and Their Graphs... Section P. Review of Trigonometric Functions... Review Eercises... Problem Solving Cengage Learning. All Rights Reserved. Ma not be scanned, copied or duplicated, or posted to a publicl accessible website, in whole or in part.

5 CHAPTER P Preparation for Calculus Section P. Graphs and Models. To find the -intercepts of the graph of an equation, let be zero and solve the equation for. To find the -intercepts of the graph of an equation, let be zero and solve the equation for.. Substitute the - and -values of the ordered pair into both equations. If the ordered pair satisfies both equations, then the ordered pair is a point of intersection.. + -intercept: (, 0 ) -intercept: ( 0, ) Matches graph (b).. 9 -intercepts: (, 0 ), (, 0) -intercept: ( 0, ) Matches graph (d).. -intercepts: (, 0 ), (, 0) -intercept: ( 0, ) Matches graph (a).. -intercepts: ( 0,0, ) (,0, ) (,0) -intercept: ( 0, 0 ) Matches graph (c) (0, ) (, ) (, ) (, ) (, 7) (0, ) (, ) (, ) (, ) (, 0 (, ) 0. ( ) (, 0) (, ) (, ) (0, ) ( (, 0) (0, 9) (, ) (, ) (, ) (, 0) (, 9) (, ) (, 0) 0 Cengage Learning. All Rights Reserved. Ma not be scanned, copied or duplicated, or posted to a publicl accessible website, in whole or in part.

6 Section P. Graphs and Models (, ) (, ) (, 0) (, ) (, ) (, ) (0, ).. 0 Undef. (, ) (, ) (, (, ) (, ) (, ( ( (, ) (, ) (, ) (, ) (, 0) (, 0) (0, ) Undef. (, ), (, ) (, ) 7. 0,, (.00, ) (,.7) (9, ) (, ) (, ) (, ) (0, ) (a) (, ) (,.7) (.7) (b) (,) (,) ( ( ) ). ( 0.,.7) (, ) (7, ) (, ) (0, ) (, ) 9 (, ) 9 (a) ( 0., ) ( 0.,.7) (b) (, ) (., ) and (, ) (, ) (, 0) Cengage Learning. All Rights Reserved. Ma not be scanned, copied or duplicated, or posted to a publicl accessible website, in whole or in part.

7 Chapter P Preparation for Calculus 9. -intercept: 0 ; ( 0, ) -intercept: 0 ;, intercept: 0 + ; ( 0,) -intercept: 0 + None. cannot equal intercept: + ; 0, 0 0 -intercepts: ( + )( ), ;, 0,, 0. -intercept: 0 ( 0) 0; ( 0, 0) -intercepts: 0 0 ( )( + ) 0, ± ; 0, 0, ±, 0. -intercept: 0 0 0; ( 0, 0) -intercepts: ,, ; 0, 0,, 0,, 0 -intercept: ; ( 0, ) + -intercept: ( ) ; (, 0) intercept: ; 0, 0 + -intercept: ;,0 + ( + ) -intercept: -intercepts: ; 0, intercept: + ( + ) ( + ) ( + ) 0, ; ( 0, 0 ), (, 0) intercept:. + -intercept: ; ( 0, ) 0; 0, ; 0, intercept: ± ;, 0 Note: is an etraneous solution. 0 Cengage Learning. All Rights Reserved. Ma not be scanned, copied or duplicated, or posted to a publicl accessible website, in whole or in part.

8 Section P. Graphs and Models 9. Smmetric with respect to the -ais because No smmetr with respect to either ais or the origin.. Smmetric with respect to the -ais because.. Smmetric with respect to the origin because ( ) ( ) + ( ) +.. Smmetric with respect to the origin because.. Smmetric with respect to the -ais because No smmetr with respect to either ais or the origin.. Smmetric with respect to the origin because Smmetric with respect to the origin because ( ) Smmetric with respect to the origin because ( ) ( ) is smmetric with respect to the -ais because. 0, -intercept 0, -intercept Intercepts: ( 0, ), (, 0 ) Smmetr: none , -intercept 0 +, -intercept Intercepts: ( 0, ), (, 0) Smmetr: none , -intercept ± 0 9 9, -intercepts Intercepts: ( 0, 9 ), (, 0 ), (, 0) 9 9 Smmetr: -ais. + ( + ) ( ) , -intercept 0 + 0,, -intercepts Intercepts: ( 0, 0 ), (, 0) Smmetr: none, 0 (, 0) (0, ) (, 0 ( (0, ) (0, 9) (, 0) 0 0. is smmetric with respect to the -ais because, 0 (0, 0). 0 Cengage Learning. All Rights Reserved. Ma not be scanned, copied or duplicated, or posted to a publicl accessible website, in whole or in part.

9 Chapter P Preparation for Calculus , -intercept 0 +, -intercept Intercepts: (, 0 ), ( 0, ) Smmetr: none (, 0) , -intercept 0 0 ( ) + 0 0, ±, -intercepts Intercepts: ( 0, 0 ), (, 0 ), (, 0) ( ) + Smmetr: origin , -intercept + 0 0,, -intercepts Intercepts: ( 0, 0 ), (, 0) Smmetr: none (0, ) (, 0) (0, 0) (, 0). 0, -intercept ±, -intercept Intercepts: ( 0, ), (, 0 ), (, 0) Smmetr: -ais (, 0) , -intercept 0, -intercept Intercept: (0, 0) Smmetr: origin ( )( ) ( )( )( ) ±, -intercepts (0, ) (, 0) 0, -intercept Intercepts: ( 0, ), ( 0, ), (, 0) Smmetr: -ais because (0, ) (0, 0) (, 0) (0, 0) (, 0) 0 (0, ) 0 Cengage Learning. All Rights Reserved. Ma not be scanned, copied or duplicated, or posted to a publicl accessible website, in whole or in part.

10 Section P. Graphs and Models 7. 0 Undefined no -intercept 0 No solution no -intercept Intercepts: none Smmetr: origin. 0, -intercept 0 0, -intercept Intercepts: (0, ), (, 0) Smmetr: none (0, ) (, 0) , -intercept No solution no -intercepts Intercept: (0, 0) 0 0 ( ) + + Smmetr: -ais. 0, -intercept 0 ±, -intercepts Intercepts: ( 0, ), (, 0 ), (, 0) Smmetr: -ais (, 0) (0, ) (, 0) 0 (0, 0) ± + ± 0 + ±, -intercepts ± , -intercept Intercepts: ( 0, ), ( 0, ), ( 9, 0) 9 Smmetr: -ais. + ± ± 0 ± ±, -intercepts + 0 ±, -intercepts Intercepts: (, 0 ), (, 0 ), ( 0, ), ( 0, ) + + Smmetr: origin and both aes (, 0) (0, ) (, 0) (0, ) ( 9, 0) 0 ( 0, ) ( 0, ) 0 Cengage Learning. All Rights Reserved. Ma not be scanned, copied or duplicated, or posted to a publicl accessible website, in whole or in part.

11 Chapter P Preparation for Calculus The corresponding -value is. Point of intersection: (, ) The corresponding -value is. Point of intersection: (, ) ( + )( ), + + The corresponding -values are ( for ) and ( for ). Points of intersection: (, ), (, ) ( ) or The corresponding -values are ( for ) and ( for ). Points of intersection: (, ), (, ). + ( ) + + or 0 The corresponding -values are and for. Points of intersection: (, ), (, ) , 0 ( ) Points of intersection: (, 0 ), (, ). + + Points of intersection: (, ), ( 0, ), (, ) ( for ) Analticall, ( )( + ) 0, 0,.. + Points of intersection: (, 0 ), ( 0, ), (, 0) Analticall, , 0,. + (, ) (0, ) (, ) + + (, 0) (, 0) (0, ) 0 Cengage Learning. All Rights Reserved. Ma not be scanned, copied or duplicated, or posted to a publicl accessible website, in whole or in part.

12 Section P. Graphs and Models (, ). (a) Using a graphing utilit, ou obtain (b) t + t (, ) 7 Points of intersection: (, ), (, ) (,.7) Analticall, ,. ( )( ) (, ) (, ) + Points of intersection: (, ), (, ) Analticall, + or or. 7. (a) Using a graphing utilit, ou obtain 0.t (b) 0 0 The model is a good fit for the data. (c) For 0, t : The GDP in 0 will be approimatel $. trillion. 9. The model is a good fit for the data. (c) For 0, t : There will be approimatel million cell phone subscribers in 0. C R To break even, 0 units must be sold. 70. k 0 0 (a) (, ): k k k (b) (, ): k k k (c) ( 0, 0 ): 0 k( 0) k can be an real number. (d) (, ): () k() 9 k k 9 7. Answers ma var. Sample answer: ( )( ) + has intercepts at,, and. 0 Cengage Learning. All Rights Reserved. Ma not be scanned, copied or duplicated, or posted to a publicl accessible website, in whole or in part.

13 0 Chapter P Preparation for Calculus 7. Yes. If (, ) is on the graph, then so is (, ) -ais smmetr. Because (, ) so is (, ) is on the graph, then b -ais smmetr. So, the graph is smmetric with respect to the origin. The converse is not true. For eample, has origin smmetr but is not smmetric with respect to either the -ais or the -ais. b 7. Yes. Assume that the graph has -ais and origin, b smmetr. If (, ) is on the graph, so is -ais smmetr. Because (, ) then so is (, ) (, ) is on the graph, b origin smmetr. Therefore, the graph is smmetric with respect to the -ais. The argument is similar for -ais and origin smmetr. 7. (a) Intercepts for : ( 0, 0 ), (, 0)(, 0) -intercept: ; 0, 0 -intercepts: 0 + ; -intercept: 0 + ; ( 0, ) -intercepts: 0 + None. cannot equal 0. Intercepts for + : (b) Smmetr with respect to the origin for because. + Smmetr with respect to the -ais for + because + +. (c) ( )( ) Point of intersection : (, ) Note: The polnomial + + has no real roots. 7. False. -ais smmetr means that if (, ) is on the graph, then (, ) is also on the graph. For eample, (, ) (, ) is on the graph. is not on the graph of 9, whereas 7. True. f f( ). b ± b ac 77. True. The -intercepts are,0. a b 7. True. The -intercept is,0. a Section P. Linear Models and Rates of Change. In the form m + b, m is the slope and b is the -intercept.. No. Perpendicular lines have slopes that are negative reciprocals of each other. So, one line has a positive slope and the other line has a negative slope.. m. m 0. m. m 0 Cengage Learning. All Rights Reserved. Ma not be scanned, copied or duplicated, or posted to a publicl accessible website, in whole or in part.

14 Section P. Linear Models and Rates of Change 7. m (, ) 7 (, ). m (, ) (, ). 0 m 0 (, ) (0, 0) 9. m, undefined. 0 The line is vertical. 7 (, ) (, ).. m 7 m m 7,, m is undefined. (, ) m m The line is horizontal. 0. m (, ) m m 0 (, ) (, ) m. Because the slope is 0, the line is horizontal and its equation is. Therefore, three additional points are ( 0,, ) (,, ) (,. ). Because the slope is undefined, the line is vertical and its equation is. Therefore, three additional points are (, 0 ), (, ), (, ). 0 Cengage Learning. All Rights Reserved. Ma not be scanned, copied or duplicated, or posted to a publicl accessible website, in whole or in part.

15 Chapter P Preparation for Calculus 7. The equation of this line is 7 ( ) + 0. Therefore, three additional points are (0, 0), (, ), and (, ).. + ( ) The equation of this line is + ( + ) +. Therefore, three additional points are (, ), ( ) and (0, )., 0, (, ) ( ) + + ( ) Because the slope is undefined, the line is vertical and its equation is.. 0 (, ) (0, ) (, ). ( + ) (, ) Since the grade of the road is, if ou drive 00 feet, 00 the vertical rise in the road will be feet.. (a) Slope Δ Δ (b) B the Pthagorean Theorem, 0 ft 0 ft feet. (0, ) 0 Cengage Learning. All Rights Reserved. Ma not be scanned, copied or duplicated, or posted to a publicl accessible website, in whole or in part.

16 Section P. Linear Models and Rates of Change 7. (a). (a) Population (in millions) Slopes: The population increased least rapidl from 009 to (b). million people per ear 9 (c) For 0, t : P P.( ) The population of the United States in 0 will be about. million people. Biodiesel production (thousands of parcels per da) Year (9 009) Year (7 007) Slopes: t The population increased most rapidl from 00 to 0. (b). thousand barrels per da 7 (c) No. The production seems to randoml increase and decrease. t 9. The slope is m and the -intercept is ( 0, ). The slope is m and the -intercept is (0, ) The slope is m and the -intercept is (0, 0). The slope is m and the -intercept is ( 0, ).. The line is vertical. Therefore, the slope is undefined and there is no -intercept.. The line is horizontal. Therefore, the slope is m 0 and the -intercept is ( 0, ) Cengage Learning. All Rights Reserved. Ma not be scanned, copied or duplicated, or posted to a publicl accessible website, in whole or in part.

17 Chapter P Preparation for Calculus. (0, ). 7 9 m ( ) + ( + ) (, 7) (, ) 9. ( ) +. 0 m ( ) 9 7 (, ) (, 0) ( + ) +. m ( ) (, ) 7 (, ) m, undefined 0 The line is vertical. or 0 0. m (, ) (, ) (, ) (, ). m 0 ( ) ( 0) + 0 (0, ) (, ) 9. m 0 The line is horizontal. or 0 (, ) (, ) 0 0 Cengage Learning. All Rights Reserved. Ma not be scanned, copied or duplicated, or posted to a publicl accessible website, in whole or in part.

18 Section P. Linear Models and Rates of Change 7 0. m, undefined 0 The line is vertical. or 0. The slope is b b. 0. The -intercept is ( 0, b ). Hence, b m + b + b. b m a b + b a 7 (0, b) 7 (, 7) (, ). ( ) ( ) a + a a + a + a a + ( ) 0 7. The given line is vertical. (a) 7, or (b), or + 0 b + a b + a b (a, 0). The given line is horizontal. (a) 0 (b), or a a 9 + a a 9 a a a m (a) ( + ) (b) ( + ) m (a) ( ) + 0 (b) ( ) Cengage Learning. All Rights Reserved. Ma not be scanned, copied or duplicated, or posted to a publicl accessible website, in whole or in part.

19 Chapter P Preparation for Calculus. 0 m (a) 7 ( ) (b) 7 ( ) m (a) (b) The slope is 0. V 0 when t. V 0( t ) + 0 0t + 0. The slope is 00.. V 7,00 when t. V 00( t ) + 7,00 00t +,00 m m m 0 ( ) 0 ( ) m The points are not collinear.. m 7. m m m The points are not collinear. (, 0) The four sides are of equal length:. For eample, the length of segment AB is ( ) units. Furthermore, the adjacent sides are perpendicular 0 because the slope of AB is, whereas the slope of BC is 0.. a + b (a) The line is parallel to the -ais if a 0 and b 0. (b) The line is parallel to the -ais if b 0 and a 0. (c) Answers will var. Sample answer: a and b (d) The slope must be. Answers will var. Sample answer: a and b. + ( + ) + (e) a and b. B (, ) (, 0) A C D (, ) Cengage Learning. All Rights Reserved. Ma not be scanned, copied or duplicated, or posted to a publicl accessible website, in whole or in part.

20 Section P. Linear Models and Rates of Change 7 9. The tangent line is perpendicular to the line joining the point (, ) and the center (0, 0). 70. The tangent line is perpendicular to the line joining the, and the center of the circle, (, ). point (, ) (0, 0) (, ) (, ) Slope of the line joining (, ) and (0, 0) is. The equation of the tangent line is ( ) Slope of the line joining (, ) and (, ) is +. Tangent line: + ( ) 0 c The slope of the perpendicular bisector b a of this segment is ( a b ) a + b c. The midpoint of this segment is,. c So, the equation of the perpendicular bisector to this segment is c a b a + b. c a, 0 and a, 0 is 7. (a) The slope of the segment joining ( b, c) and ( a, 0) is. Similarl, the equation of the perpendicular bisector of the segment joining c a b b a. c Equating the right-hand sides of each equation, ou obtain 0. Letting 0 in either equation ields the point of intersection: c a b a + b c b a c + b a c c c c a + b + c The point of intersection is 0,. c (b) The equations of the medians are: c b c c a a b a b a a c c a a a + b + a + b + + a. Solving these equation simultaneousl for (, ), b a c, ( a, 0) (0, 0) (b, c) a + b c, (a, 0) b a c, ( a, 0) b c ou obtain the point of intersection,. (b, c) a + b c, (a, 0) 0 Cengage Learning. All Rights Reserved. Ma not be scanned, copied or duplicated, or posted to a publicl accessible website, in whole or in part.

21 Chapter P Preparation for Calculus 7. (a) Lines c, d, e and f have positive slopes. (b) Lines a and b have negative slopes. (c) Lines c and e appear parallel. Lines d and f appear parallel. (d) Lines b and f appear perpendicular. Lines b and d appear perpendicular. 7. Find the equation of the line through the points (0, ) and (00, ) C m ( C ) F 0 F + or C 9 0 F 9C 0 0 For F 7, C.. ( F ) 7. (a) Current job: W 0.07s (b) New job offer: W 0.0s (,000, 00) 0 00 Using a graphing utilit, the point of intersection is (,000, 00). Analticall, W W 0.07s s s 00 s,000 0,000 So, W W 0.07, When sales eceed $,000, the current job pas more. (c) No, if ou can sell $0,000 worth of goods, thenw > W. (Note: W 00 and W 00 when s 0,000.) 7. (a) Two points are (0, 70) and (7, ). The slope is 70 m. 7 0 p 70 ( 0) p or 0 (b) ( p) (a) (b) ( quiz score, test score) (c) If 7, (d) The slope shows the average increase in eam score for each unit increase in quiz score. (e) The points would shift verticall upward units. The new regression line would have a -intercept greater than before: If p, then units. (c) If 79, units p then 0 Cengage Learning. All Rights Reserved. Ma not be scanned, copied or duplicated, or posted to a publicl accessible website, in whole or in part.

22 Section P. Linear Models and Rates of Change If A 0, then B + C 0 is the horizontal line C B. C B + C A + B + C d. B B A + B If B 0, then A + C 0 is the vertical line C A. The distance to (, ) is The distance to (, ) is d C A + C A + B + C. A A A + B (Note that A and B cannot both be zero.) The slope of the line A + B + C 0 is A B. The equation of the line through (, ) perpendicular to A + B + C 0 is: B ( ) A A A B B B A B A The point of intersection of these two lines is: A + B C A+ AB AC B A B A B AB B AB A + B AC + B AB () ( B adding equations () and ) AC + B AB A + B A + B C AB + B BC () B A B A AB + A AB + A ( A + B ) BC AB + A ( B adding equations () and () ) BC AB + A A + B AC + B AB BC AB + A, point of intersection A + B A + B,, and the line A + B + C 0. The distance between ( ) and this point gives ou the distance between ( ) AC + B AB BC AB + A d + A + B A + B AC AB A BC AB B + A + B A + B A C + B + A B C + A + B A B C A B A B C + A + B A + B + A + B m + m A + B + C m + + m + d A + B m + m + ( A B ) The distance is 0 when m. In this case, the line + contains the point (, ). 9 9 (, 0) 0 Cengage Learning. All Rights Reserved. Ma not be scanned, copied or duplicated, or posted to a publicl accessible website, in whole or in part.

23 0 Chapter P Preparation for Calculus d d +. For simplicit, let the vertices of the rhombus be (0, 0), (a, 0), (b, c), and ( a + b, c), as shown in the figure. c The slopes of the diagonals are then m and a + b c m. Because the sides of the rhombus are b a equal, a b + c, and ou have mm c c c c a + b b a b a c Therefore, the diagonals are perpendicular.. (0, 0) (b, c) (a + b, c) (a, 0). For simplicit, let the vertices of the quadrilateral be (0, 0), (a, 0), (b, c), and (d, e), as shown in the figure. The midpoints \of the sides are a a + b c b + d c + e d e,0,,,,,and,. The slope of the opposite sides are equal: c c + e e 0 c a + b a b + d d b e c c + e 0 e a d a + b b + d a d Therefore, the figure is a parallelogram. d e, (d, e) b + d c e, + (b, c) a + b c, (0, 0) a, 0 (a, 0). Consider the figure below in which the four points are collinear. Because the triangles are similar, the result immediatel follows. * * * * (, ) (, ) ( *, * ) ( *, * ). If m m, then mm. Let L be a line with slope m that is perpendicular to L. Then mm. So, m m L and L are parallel. Therefore, L and L are also perpendicular.. True. a c a a + b c + m b b b b c b b a c m a a a m m. True. The slope must be positive. 0 Cengage Learning. All Rights Reserved. Ma not be scanned, copied or duplicated, or posted to a publicl accessible website, in whole or in part.

24 Section P. Functions and Their Graphs Section P. Functions and Their Graphs. A relation between two sets X and Y is a set of ordered pairs of the form (, ), where is a member of X and is at member of Y. A function from X to Y is a relation between X and Y that has the propert that an two ordered pairs with the same -value also have the same -value.. For a function f from X to Y, the domain is the set X. If is the image of, then the range is a subject of Y consisting of all images of numbers in X.. The three basic tpes are vertical shifts, horizontal shifts, and reflections.. Consider the nonconstant polnomial n n f a + a + + a + a n n 0. If n is even and a n > 0, then the graph of f moves up to the left and up to the right. If n is even and a n < 0, then the graph of f moves down to the left and down to the right. If n is odd and a n > 0, then the graph of f moves down to the left and up to the right. If n is odd and a n < 0, then the graph of f moves up to the left and down to the right.. f (a) f ( 0) ( 0) (b) f ( ) ( ) (c) f( b) ( b) b (d) f. (a) f ( 0) 7( 0) (b) f ( ) 7( ) (c) f( b) 7( b) 7b (d) f (a) f ( ) ( ) + + (b) f () (c) f + + (d) f( b) ( b) b + b +. (a) f ( ) + (b) f + (c) f + 9 (d) f( +Δ ) +Δ + 9. (a) g ( 0) 0 (b) g ( ) ( ) 0 (c) g( ) ( ) (d) gt ( ) ( t ) ( t t+ ) + t t 0. (a) g 0 (b) 9 g (c) g( c) c ( c ) c c (d) gt ( + ) ( t+ ) ( t+ ) t + t t + t + t.. f +Δ f +Δ + Δ + Δ + Δ + Δ + ( Δ ) Δ Δ Δ Δ ( ) ( ) f f,, 0. f Domain: (, ) Range: [ 0, ). g Domain: (, ) Range: [, ) 0 Cengage Learning. All Rights Reserved. Ma not be scanned, copied or duplicated, or posted to a publicl accessible website, in whole or in part.

25 Chapter P Preparation for Calculus. f ( ) ( ) Domain:, Range:,. h Domain:, Range:, 7. g ( ) ( ] Domain: 0 0 0, Range: [ 0, ). h + [ ) Domain: + 0 [, ) Range: (,0] Domain:, Range: 0, 9. f 0 [ ] [ ] Note: is a semicircle of radius. 0. f Domain:, Range: 0,. f ( ) [ ) Domain: all 0 (, 0) ( 0, ) Range: (,0) ( 0, ) + Domain: all Range: all. f [Note: You can see that the range is all b graphing f.]. f + 0 and 0 0 and 0 0, Domain: [ ]. f ( )( ) Domain: or Domain: (,] [, ). f Domain: all Domain: (, ) (, ). g 0 ( )( ) + 0 Domain: all ± Domain: (, ) (, ) (, ) 7. f +, < 0 +, 0 (a) f ( ) ( ) + (b) f ( 0) ( 0) + (c) f + (d) f t + t + + t + (Note: t + 0for all t.) Domain: (, ) Range: (,) [, ). f +, +, > (a) f ( ) ( ) + (b) f ( 0) 0 + (c) f () + (d) f s + s + + s + s + 0 (Note: s + > for all s.) Domain: (, ) Range: [, ) 0 Cengage Learning. All Rights Reserved. Ma not be scanned, copied or duplicated, or posted to a publicl accessible website, in whole or in part.

26 Section P. Functions and Their Graphs 9. f +, < +, (a) f ( ) + (b) f () + 0 (c) f () + (d) f b + b + + b Domain: (, ) Range: (, 0] [, ) g. + Domain: (, ) Range: ( 0, 0. f +, ( ), > (a) f ( ) + (b) f ( 0) 0 + (c) f ( ) + (d) f ( 0) ( 0 ) Domain: [, ) Range: [ 0, ). f Domain: (, ) Range: (, ). f() t 7 + t Domain: (, 7) ( 7, ) Range: (, 0) ( 0, ). h Domain: 0, Range: [ 0, ) t [ ). f + Domain: (, ) Range: [, ) 0 9. f + Domain: (, ) Range: (, ) 0 Cengage Learning. All Rights Reserved. Ma not be scanned, copied or duplicated, or posted to a publicl accessible website, in whole or in part.

27 Chapter P Preparation for Calculus f 9 7. Domain: [, ] Range: [ 0, ]. + is a function of because there is one value of for each... ± is not a function of because there are two values of for some is a function of because there is one value of for each. f Domain: [, ] Range:, [,.] -intercept: ( 0, ) -intercept: (, 0) (, 0 0 ± is not a function of. Some vertical lines intersect the graph twice is a function of. Vertical lines intersect the graph at most once.. is a function of. Vertical lines intersect the graph at most once.. + ( (0, ) ± is not a function of. Some vertical lines intersect the graph twice.. + ± is not a function of because there are two values of for some. 7. The transformation is a horizontal shift two units to the f right of the function. Shifted function:. The transformation is a vertical shift units upward of f the function. Shifted function: + 9. The transformation is a horizontal shift units to the right and a vertical shift unit downward of the function f. Shifted function: ( ) 0. The transformation is a horizontal shift unit to the left and a vertical shift units upward of the function f. Shifted function: ( ). f( ) is a horizontal shift units to the left. Matches d.. f is a vertical shift units downward. Matches b.. f( ) is a reflection in the -ais, a reflection in the -ais, and a vertical shift downward units. Matches c.. f( ) is a horizontal shift units to the right, followed b a reflection in the -ais. Matches a.. f( ) + + is a horizontal shift to the left units, and a vertical shift upward units. Matches e.. f( ) + is a horizontal shift to the right unit, and a vertical shift upward units. Matches g. 0 Cengage Learning. All Rights Reserved. Ma not be scanned, copied or duplicated, or posted to a publicl accessible website, in whole or in part.

28 Section P. Functions and Their Graphs 7. (a) The graph is shifted units to the left. (f ) The graph is stretched verticall b a factor of. (b) The graph is shifted unit to the right. (g) The graph is a reflection in the -ais. (c) The graph is shifted units upward. (h) The graph is a reflection about the origin. (d) The graph is shifted units downward. (e) The graph is stretched verticall b a factor of. 0 0 Cengage Learning. All Rights Reserved. Ma not be scanned, copied or duplicated, or posted to a publicl accessible website, in whole or in part.

29 Chapter P Preparation for Calculus. (a) g f( ) g( ) f g f 0 The graph is shifted units to the right. (, ) 7 (0, ) (b) g f( + ) g( 0) f g f The graph is shifted units to the left. (0, ) 7 (, ) (c) g f g f g f The graph is shifted units upward. (, ) (, ) (d) g f g f g f 0 The graph is shifted unit downward. (, 0) (e) g f g f g f The graph is stretched verticall b a factor of. (, ) (, ) (f ) g f g f ( ) f g The graph is stretched verticall b a factor of., (g) g f( ) g f g f The graph is a reflection in the -ais. (h) g f g f g f, (, ) (, ) The graph is a reflection in the -ais. (, ) (, ) (, ) 0 Cengage Learning. All Rights Reserved. Ma not be scanned, copied or duplicated, or posted to a publicl accessible website, in whole or in part.

30 Section P. Functions and Their Graphs 7 9. f, g (a) f + g ( ) + ( ) (b) f g ( ) ( ) 9 (c) (d) f g f g f + +, g + f g (a) (b) f g (c) (d) f g f g +, + + () (c) g( f) g (d) f( g( )) f (e) ( ). (a) f g() f 0 0 (b) g f() g f g f, ( 0) g f g (f). f, g + (a) f( g) f() () 0 (b) f( g) f( ) f + 0 (c) g( f) g 0 0 (d) ( ) g f g g( ) ( ) + (e) ( ) (f) g( f ) g( ) ( ) + + f g f f, g ( f g) f g f Domain: [ 0, ) ( g f) g( f ) g( ) Domain: (, ) No. Their domains are different. ( f g) ( g f), 0 for 0. 0 Cengage Learning. All Rights Reserved. Ma not be scanned, copied or duplicated, or posted to a publicl accessible website, in whole or in part.

31 Chapter P Preparation for Calculus. f, g ( ) f g f g f Domain: (, ) ( ) g f g Domain: (, ) f g g f f, g. ( ) f g f g f Domain: all ± (, ) (, ) (, ) ( ) ( ) 9 9 g f g f g Domain: all 0 (, 0) ( 0, ) f g g f. ( f g) f( ) Domain: (, ) ( g f) g + You can find the domain of g f b determining the intervals where ( + ) and are both positive, or both negative Domain: (,, ( 0, ) 7. (a) ( f g)() f( g() ) f( ) (b) g( f ) g() (c) g( f( ) ) g( ), (d) ( f g) f( g) f (e) ( g f) g( f) g (f) f( g ) f( ), 9. F Let h, g and f. Then ( f g h) f( g ) f( ) F. (Other answers possible.) F f, g, and h. Let Then ( f g h) f( g( )) f( ). (Other answers possible.) which is undefined which is undefined. () ( ()) A r t A r t A 0.t 0.t 0.t ( A r)( t) represents the area of the circle at time t. 7. (a) If f is even, then ( ) (b) If f is odd, then ( ), is on the graph., is on the graph. 7. (a) If f is even, then (, 9) is on the graph. (b) If f is odd, then (, 9) is on the graph. 7. f is even because the graph is smmetric about the -ais. g is neither even nor odd. h is odd because the graph is smmetric about the origin. 0 Cengage Learning. All Rights Reserved. Ma not be scanned, copied or duplicated, or posted to a publicl accessible website, in whole or in part.

32 Section P. Functions and Their Graphs 9 7. (a) If f is even, then the graph is smmetric about the -ais. (b) If f is odd, then the graph is smmetric about the origin. 7. f ( ) f is even. f Zeros: 0,, ( ) ( ) f f 7. f ( ) ( ) f f f is odd. f 0 0 is the zero. 77. f f The domain of f is 0 and the range is 0. Hence, the function is neither even nor odd. The onl zero is f f ( ) ( ) ( ) f f is even. 0 Zeros: 0, ± f Slope 0 ( ( ) ) 0 For the line segment, ou must restrict the domain., f 0 0. Slope 7 7 ( ) For the line segment, ou must restrict the domain. f 7 9,. (, ) + 0 f, 0 (0, ) (, ) (, ) 0 Cengage Learning. All Rights Reserved. Ma not be scanned, copied or duplicated, or posted to a publicl accessible website, in whole or in part.

33 0 Chapter P Preparation for Calculus. +,. Answers will var. Sample answer: The graph begins at time 0, then decreases until ear. The graph then increases slightl for a few ears and then decreases again. Value (in thousands of dollars) 9 Time (in ears) t. Answers will var. Sample answer: Speed begins and ends at 0. The speed might be constant in the middle: Speed (in miles per hour). Answers will var. Sample answer: Height begins a few feet above 0, and ends at 0. Height Time (in hours) Distance. Answers will var. Sample answer: Distance begins at 0, then the graph has a sharp turn after a few minutes and goes back to 0. Then the graph goes back upward steepl. Distance from home (in miles) 9 0 Time (in minutes) t 7. c c + c, a circle. For the domain to be [, ], c.. For the domain to be the set of all real numbers, ou must require that + c + 0. So, the discriminant must be less than zero: ( c) < 0 c < 9 c < < c < c < < 9. No. If a horizontal line intersects the graph more than once, then there is more than one -value corresponding to the same -value. 90. Answers will var. Sample answer: f and g ( f g) f ( g f) g 9. No. For eample, + + is not odd since f ( ) f. 9. f( ) 0 is even and odd. f ( ) 0 f f( ) 0 Cengage Learning. All Rights Reserved. Ma not be scanned, copied or duplicated, or posted to a publicl accessible website, in whole or in part.

34 Section P. Functions and Their Graphs 9. (a) T T (b) If Ht () Tt,, then the changes in temperature will occur hour later. (c) If Ht () Tt (), then the overall temperature would be degree lower. 9. (a) For each time t, there corresponds a depth d. (b) Domain: 0 t Range: 0 d 0 (c) d (d) d. At time seconds, the depth is approimatel cm. t 9. (a) (b) H p + has one zero. p has three zeros. Ever cubic polnomial has at least one zero. Given p A + B + C + D, ou have p as and p as if A > 0. Furthermore, p as and p as if A < 0. Because the graph has no breaks, the graph must cross the -ais at least one time. n+ 97. f ( ) a ( ) + + a ( ) + a ( ) Odd n+ n+ an+ a a f n n 9. n n 0 n n n n 0 f a + a + + a + a a + a + + a + a f Even 99. Let F f g where f and g are even. Then F( ) f( ) g( ) f g F. So, F is even. Let F f g where f and g are odd. Then F( ) f( g ) ( ) f g f( g ) F. So, F is even. 00. Let F f g where f is even and g is odd. Then F( ) f( ) g( ) f g f g F. So, F is odd. 0. B equating slopes, 0 0 +, L Cengage Learning. All Rights Reserved. Ma not be scanned, copied or duplicated, or posted to a publicl accessible website, in whole or in part.

35 Chapter P Preparation for Calculus V 0. (a) (b) Domain: 0 < < First consider the portion of R in the first quadrant: 0, 0 and ; shown below. (0, ) (, ) Maimum volume occurs at. So, the dimensions of the bo would be cm. 9, but 0. False. If f, then f f. 0. True 00 (0, 0) (, 0) The area of this region is +. B smmetr, ou obtain the entire region R: (, ) (, ) 0. True. The function is even. f 9 and 0. False. If f then, So, f f( ). f. 07. False. The constant function f( ) 0 has smmetr with respect to the -ais. 0. True. If the domain is { }, { }. a then the range is f ( a ) (, ) The area of R is 0. Let g (, ). cbe constant polnomial. Then f ( g ) fc () and g( f ) c. So, f () c c. Because this is true for all real numbers c, f is the identit function: f. Section P. Review of Trigonometric Functions. In general, if θ is an angle measured in degrees, then the angle θ + n( 0 ), n a nonzero integer, is coterminal with θ.. If the degree measure of angle θ is, then the radian measure of θ is. 0. sin θ opp 7 hp cos θ adj hp tan θ opp 7 adj. The amplitude of the graph of a sin b or a cos bis a. This value is the maimum value of the function, and a is the minimum value. A function is periodic if there eists a positive real number p such that f ( + p) f for all in the domain of f. The period of f is the least positive value of p.. (a) θ θ 0 0 (b) θ θ Cengage Learning. All Rights Reserved. Ma not be scanned, copied or duplicated, or posted to a publicl accessible website, in whole or in part.

36 Section P. Review of Trigonometric Functions. (a) θ θ (b) θ 7. (a) (b). (a) (b) θ θ θ θ θ 9 θ θ + + θ θ (a) (b) (c) (d) (a) (b) (c) (d) (a) 0 70 (b) (c) (d) (a) (b) 0 0 (c) 0 0 (d). (a). (a) r ft in. cm in. s ft in. 9 in. mi mi h 00 ft min 0 Circumference of tire: C. feet Revolutions per minute: radians. (b) θ ( ) θ 0 radians Angular speed: 0 rad min t minute,, r csc θ sec θ cot θ sin θ cos θ tan θ (b),, r csc θ sec θ cot θ sin θ cos θ tan θ 0 Cengage Learning. All Rights Reserved. Ma not be scanned, copied or duplicated, or posted to a publicl accessible website, in whole or in part.

37 Chapter P Preparation for Calculus. (a),, r 7 7 sin θ csc θ 7 7 cos θ sec θ 7 tan θ cot θ (b),, r sin θ csc θ cos θ sec θ tan θ cot θ + cos θ 7. θ. (a) (b) (c) (d) sin 0 cos 0 tan 0 sin 0 sin 0 cos 0 cos 0 tan 0 tan 0 sin cos tan sin sin cos cos tan tan. + tan θ + 9. cot θ θ + 0. csc θ θ sin 0 sin 0 cos( 0 ) cos 0 tan( 0 ) tan 0. (a) (b) (c) sin 0 sin 0 cos 0 cos 0 tan 0 tan 0 sin sin cos cos tan tan θ (d) sin cos 0 tan is undefined. 0 Cengage Learning. All Rights Reserved. Ma not be scanned, copied or duplicated, or posted to a publicl accessible website, in whole or in part.

38 Section P. Review of Trigonometric Functions. (a) sin sin cos cos tan tan sin sin (b) cos( ) cos tan tan (c) sin sin cos cos tan tan (d) sin sin cos cos tan tan. (a) sin 70 sin 0 cos 70 cos 0 tan 70 tan 0 (b) sin 0 sin 0 cos 0 cos 0 tan 0 tan 0 0 (c) sin sin 0 cos cos 0 tan tan 7 (d) sin sin 7 cos cos 7 tan tan. (a) sin (b) csc (a) sec. (b) sec. 7. (a) tan (b) 0 tan (a) cot. 0. (b) tan.. 9. (a) sin θ < 0 θ is in Quadrant III or IV. cos θ < 0 θ is in Quadrant II or III. sin θ < 0 and cos θ < 0 θ is in Quadrant III. (b) sec θ > 0 θ is in Quadrant I or IV. cot θ < 0 θ is in Quadrant II or IV. sec θ > 0 and cot θ < 0 θ is in Quadrant IV. 0. (a) sin θ > 0 θ is in Quadrant I or II. cos θ < 0 θ is in Quadrant II or III. sin θ > 0 and cos θ < 0 θ is in Quadrant II. (b) csc θ < 0 θ is in Quadrant III or IV. tan θ > 0 θ is in Quadrant I or III. csc θ < 0 and tan θ > 0 θ is in Quadrant III.. (a) (b) cos θ θ 7, cos θ θ,. (a) sec θ θ, (b) sec θ θ,. (a) tan θ θ, (b) cot θ θ, 0 Cengage Learning. All Rights Reserved. Ma not be scanned, copied or duplicated, or posted to a publicl accessible website, in whole or in part.

39 Chapter P Preparation for Calculus. (a) (b) sin θ θ, sin θ θ,. θ θ cos + sin θ + θ sin sin θ θ sin sin 0 ( θ ) sin θ sin 0 sin θ 0 sin θ θ 0,, θ. θ sin θ ± 7 θ,,, sin. tan θ tan θ ± θ,,, tan θ tan θ 0 tan θ tan 0 ( θ ) tan θ 0 tan θ θ 0,, θ, cos θ cos θ 0 ( θ )( θ ) cos + cos 0 cos θ cos θ θ, θ 0, sec θ csc θ csc θ 0 csc θ( sec θ ) 0 ( csc θ / 0 for an value of θ) 0. sin θ cos θ tan θ θ, sec θ θ, θ. cos cos θ θ cos cos θ + ( + cos θ) cos θ + cos θ cos θ θ θ ( 0 is etraneous).. cos cos + θ θ + cos θ + 7 ft sec 0 sec,00 feet θ + θ + 0 cos cos 0 cos + cos + a sin,00 a,00 sin 099 feet,00 ft h h tan. and tan tan. h tan 9 h tan. + tan. tan 9 ( θ θ ) 0 cos + cos + ( θ )( θ ) tan. tan 9 tan. tan. tan 9 tan. h tan. tan 9 tan 9 tan 9 tan..9 mi or 9.07 ft 0 Cengage Learning. All Rights Reserved. Ma not be scanned, copied or duplicated, or posted to a publicl accessible website, in whole or in part.

40 Section P. Review of Trigonometric Functions 7. sin Period Amplitude. cos Period Amplitude 7. sin Period Amplitude. cos 0 Period 0 0 Amplitude 9. tan Period 0. 7 tan Period. sec Period. csc Period. (a) f c sin ; changing c changes the amplitude. When c : f sin. When c : f sin. When c : f sin. When c : f sin. (b) f cos ( c) ; changing c changes the period. When c : f cos( ) cos. When c : f cos( ) cos. When c : f cos. When c : f cos. (c) f cos ( c) ; When c : f cos( + ). When c : f cos( + ). When c : f cos( ). When c : f cos( ). changing c causes a horizontal shift. c c c c.. c ± c ± c. c c c. 0 Cengage Learning. All Rights Reserved. Ma not be scanned, copied or duplicated, or posted to a publicl accessible website, in whole or in part.

41 Chapter P Preparation for Calculus. (a) f sin + c; changing c causes a vertical shift. When c : f sin. When c : f sin. When c : f sin +. When c : f sin +. (b) f sin( c) ; When c : f sin( + ). When c : f sin( + ). When c : f sin( ). When c : f sin( ). changing c causes a horizontal shift. (c) f c cos ; changing c changes the amplitude. When c : f cos. When c : f cos. When c : f cos. When c : f cos. c c.. c c. c 0.7. c c c c.7.7 c c c. sin Period: Amplitude: 7. sin Period: Amplitude:. cos Period: Amplitude:. tan Period: 0 Cengage Learning. All Rights Reserved. Ma not be scanned, copied or duplicated, or posted to a publicl accessible website, in whole or in part.

42 Section P. Review of Trigonometric Functions 9 9. csc Period:. sin( + ) Period: Amplitude: 0. tan Period:. cos Period: Amplitude:. sec Period:. + cos Period: Amplitude:. csc Period:. + sin + Period: Amplitude: 0 Cengage Learning. All Rights Reserved. Ma not be scanned, copied or duplicated, or posted to a publicl accessible website, in whole or in part.

43 0 Chapter P Preparation for Calculus 7. a cos( b c) From the graph, we see that the amplitude is, the period is, and the horizontal shift is. Thus, a b b c c. d Therefore, ( ). a sin( b c) cos. From the graph, we see that the amplitude is, the period is, and the horizontal shift is 0. Also, the graph is reflected about the -ais. Thus, a b b c 0 c 0. b Therefore, sin. 9. Yes. Use the right-triangle definitions of the trigonometric functions. 70. The sine function is one-to-one on the interval,. Other intervals are possible. 7. The range of the cosine function is [, ]. The range of the secant function is (, ] [, ). 7. As θ increases from 0 to 90 with r centimeters, decreases from to 0 centimeters, increases from 0 to centimeters, sinθ increases from 0 to, cosθ decreases from to 0, and tanθ increases from 0 to (positive) infinit. 7. f g h sin sin sin f() sin The graph of f ( ) will reflect an parts of the graph of f ( ) below the -ais about the -ais. The graph of f ( ) will reflect the part of the graph of f ( ) to the right of the -ais about the -ais. 7. If h + 0 sin t, then h when t 0. t 7. S. +. cos 00 g() sin h() sin( ) 0 0 Sales eceed 7,000 during the months of Januar, November, and December. 0 Cengage Learning. All Rights Reserved. Ma not be scanned, copied or duplicated, or posted to a publicl accessible website, in whole or in part.

44 Review Eercises for Chapter P f sin + sin 7. g sin + sin + sin f sin sin sin sin n, n n,, Pattern: ( ) 77. False. radians (not radians) corresponds to two complete revolutions from the initial side to the terminal side of an angle. 7. True 79. False. The amplitude of the function sin is one-half the amplitude of the function sin. 0. True Review Eercises for Chapter P. 0: 0 ( 0,, ) -intercept 0: 0 (, 0 ), -intercept.. + 0: ,, -intercept + 0: 0,, 0,, 0, -intercepts 0: 0 0,, -intercept 0 0: 0 (, 0 ), -intercept +. 0: ,, -intercept 0: + 0,,0,,0, -intercepts. + does not have smmetr with respect to either ais or the origin.. Smmetric with respect to -ais because Smmetric with respect to both aes and the origin because:. Smmetric with respect to the origin because: ( )( ). 0 Cengage Learning. All Rights Reserved. Ma not be scanned, copied or duplicated, or posted to a publicl accessible website, in whole or in part.

45 Chapter P Preparation for Calculus (0, ) -intercept: -intercept: + 0 Smmetr: none (, 0) (0, ) (, 0) (0, ) -intercepts: + 0 ( )( + ) 0 ±, 0,, 0 -intercept: Smmetric with respect to the -ais because + +. (0, ) (, 0) (, 0) ,, Intercepts: ( 0, 0, ) (, 0, ) (, 0) Smmetric with respect to the origin because ( ) 9( ) ( ) 9 + ( 9 ). f f (, 0) 9 (0, 0) (, 0) intercept: ± 0,, 0, -intercept: , 0 Smmetric with respect to the -ais because (0, ). -intercept: 0 0, -intercept: 0 0 0, 0 Smmetr: none (0, ) (9, 0) (0, ) 7 9 (, 0) 0 Cengage Learning. All Rights Reserved. Ma not be scanned, copied or duplicated, or posted to a publicl accessible website, in whole or in part.

46 Review Eercises for Chapter P. -intercept: 0 0 0, 0 -intercepts: Smmetr: none 0 or 0 ( 0, 0 ), (, 0). + ( ). + ( ) 0 (0, 0) (, 0) + + For, + +. Point of intersection is: (, ) For, 7 Point of intersection:, or ( )( ) For, +. For, +. Points of intersection: (, ), (, ) ( ) ( + ) 0or For 0, 0 +. For, + 0. Points of intersection: ( 0, ), (, 0) (, ) (, ) Slope The line is horizontal and has slope 0. ( 7, ) (, ) 0 Cengage Learning. All Rights Reserved. Ma not be scanned, copied or duplicated, or posted to a publicl accessible website, in whole or in part.

47 Chapter P Preparation for Calculus. ( ) 7 ( ) ( 0, (, ) ( 7. Slope: 0 -intercept: 0, 7. Because m is undefined the line is vertical. or + 0. Slope: undefined Line is vertical. (, ). 0 ( ( ) ) (, 0) 9. Slope: ( ) -intercept: 0,. Because m 0, the line is horizontal. ( ) 0 or Slope : -intercept: 0,. + Slope: m -intercept: ( 0, ) Slope: m (, ).. m ( 0) 0 m 0 ( ) ( ( ) ) intercept: ( 0, 9 ) 0 Cengage Learning. All Rights Reserved. Ma not be scanned, copied or duplicated, or posted to a publicl accessible website, in whole or in part.

48 Review Eercises for Chapter P. (a) 7 ( + ) (b) has slope. ( ) (c) Perpendicular line has slope. ( ( ) ) or + 9 (d) Slope is undefined so the line is vertical (a) ( ) (b) + 0 has slope. Slope of the perpendicular line is. ( ) (c) The slope of the line 0 is. The parallel line through (, ) is ( ) + 0. (d) Because the line is horizontal the slope is The slope is 0. V 0t +,00. V () 0() +,00 $990. (a) C 9.t +.0t +,00.7t +,00 (b) R 0t (c) 0t.7t +,00 7.t,00 t 0. hours to break even 7. f + (a) f + (b) f + (c) f (d) f t + t + + t + 9 f (a) (b) f 7 + f (c) f + (d) f( c ) ( c ) ( c ) 9. f c c + c c + c c + c + f +Δ f +Δ Δ Δ ( + Δ + ( Δ) ) Δ Δ + ( Δ) Δ + Δ + ( Δ) Δ + Δ, Δ 0 0. f f () () ( ) ( ) f f + ( ), 0 Cengage Learning. All Rights Reserved. Ma not be scanned, copied or duplicated, or posted to a publicl accessible website, in whole or in part.

49 Chapter P Preparation for Calculus. f + Domain:, Range:,. g ( ) [ ) Domain: 0 Range: 0,. f (,] [ ) + Domain:, Range:, 0. h ( ) ( ] + Domain: all ;,, Range: all 0;, 0 0,. f Domain: (, ) (, ) Range: (, 0) ( 0, ). g + Domain: [, ) Range: [ 0, ) 7. + ± is not a function of. Some vertical lines intersect the graph more than once is a function of. A vertical line intersects the graph eactl once. + 0 ( ) is a function of ± 9 is not a function of. Some -values correspond to the same -value.. f (0, 0) (a) The graph of g is obtained from f b a vertical shift down unit, followed b a reflection in the -ais: g f + + (b) The graph of g is obtained from f b a vertical shift upwards of and a horizontal shift of to the right. g f + +. (a) (cubic), negative leading coefficient (b) (quartic), positive leading coefficient (c) (quadratic), negative leading coefficient (d), positive leading coefficient. f +, g ( ) ( ) f g f g f + Domain: (, ) ( g f) g( f ) g( + ) ( + ) Domain: (, ) f g g f., ( ) f g Domain: (,, ) ( g f) g( f ) g( ) ( ) f g f g f f g g f (, ) 0 Cengage Learning. All Rights Reserved. Ma not be scanned, copied or duplicated, or posted to a publicl accessible website, in whole or in part.

50 Review Eercises for Chapter P 7. f f ( ) ( ) ( ) f f is even. ( ) Zeros: 0,,. f f f + + f is neither even nor odd. f Zero: sin sin. cos cos tan( ) tan sin 0 sin 0 cos 0 cos 0 tan 0 tan 0 sin sin cos cos tan tan sin sin cos cos tan tan sin 0 sin cos 0 cos tan 0 tan 70. sin 0 0 cos 0 tan tan cot 0.0 tan 0 sec. cos ( ) csc.7 9 sin 9 ( ) 7. sin cos Cengage Learning. All Rights Reserved. Ma not be scanned, copied or duplicated, or posted to a publicl accessible website, in whole or in part.

51 Chapter P Preparation for Calculus 77. cos θ + 0 cos θ θ, 7. cos θ cos θ cos θ ± 7 θ,,, sin θ + sin θ + 0 ( θ )( θ ) sin sin + 0 sin θ or sin θ 7 θ, or θ cos θ cos θ cos θ cos 0 ( θ ) θ( θ) cos sin 0 cos θ 0 or sin θ 0 θ, or θ 0,, sec θ sec θ 0 ( θ )( θ ) sec sec + 0 sec θ or sec θ cos θ or cos θ θ, or θ sec θ + tan θ 0 ( θ ) sec θ + sec 0 sec θ sec θ sec θ ± 7 θ,,,. 9 cos Period: Amplitude: 9. sin Period: Amplitude:. 0 0 sin Period: Amplitude:. cos Period: Amplitude: 0 Cengage Learning. All Rights Reserved. Ma not be scanned, copied or duplicated, or posted to a publicl accessible website, in whole or in part.

52 Problem Solving for Chapter P 9 7. tan Period: 9. sec Period:. cot Period: 90. csc Period: Problem Solving for Chapter P. (a) + 0 ( ) ( ) ( ) ( ) Center: (, ); Radius: + (b) Slope of line from (0, 0) to (, ) is. Slope of tangent line is. So, 0 ( 0 ), Tangent line (c) Slope of line from (, 0) to (, ) is 0. (d) Slope of tangent line is Intersection:, 0 9, Tangent line So, 0 Cengage Learning. All Rights Reserved. Ma not be scanned, copied or duplicated, or posted to a publicl accessible website, in whole or in part.

53 0 Chapter P Preparation for Calculus. Let m + be a tangent line to the circle from the point (0, ). Because the center of the circle is at ( 0, ) and the radius is ou have the following. + + ( m ) m m Setting the discriminant b ac equal to zero, m m + 0 () m m m m ± Tangent lines: + and +.. H (a) H, 0 0, < 0, 0, < 0 (c) H (d) H( ) (e) H (f ) H( ), 0 0, < 0, 0 0, > 0, 0 0, < 0, +, < (b) H( ), 0, < 0 Cengage Learning. All Rights Reserved. Ma not be scanned, copied or duplicated, or posted to a publicl accessible website, in whole or in part.

54 Problem Solving for Chapter P. (a) f( + ) (f ) f ( ) (b) f( ) + (g) f ( ) (c) f ( ). (a) (b) A + 0 Domain: , < < or (d) f ( ) (e) f 0 0 Maimum of 0 (c) A ( 00 ) ( ) ( ) m at 0 m, m A ( 0) 0 m is the maimum. 0 m, m 0 0 Cengage Learning. All Rights Reserved. Ma not be scanned, copied or duplicated, or posted to a publicl accessible website, in whole or in part.

55 Chapter P Preparation for Calculus 00. (a) A Domain: 0 < < 00 (b) Maimum of 70 ft at 0 ft, (c) A ( 00 ) ( ) ( ) ft. A ( 0) 70 square feet is the maimum area, where 0 ft and 7. ft. 7. The length of the trip in the water is +, and the length of the trip over land is ( ). + ( ) total time is T So, the + + hours.. Let d be the distance from the starting point to the beach. distance Average speed time d d d km h 9. (a) Slope 9. Slope of tangent line is less than. (b) Slope. Slope of tangent line is greater than. (c) Slope.... Slope of tangent line is less than.. (d) Slope f ( + h) f + h ( h) + h h + h h + h, h 0 (e) Letting h get closer and closer to 0, the slope approaches. So, the slope at (, ) is. 0. (, ) (a) Slope. Slope of tangent line is greater than. 9 (b) Slope. Slope of tangent line is less than.. 0 (c) Slope. Slope of tangent line is greater than 0.. f + h f + h (d) Slope + h h (e) ( h) + h + h + h + +, h 0 h h + h + h + h + ( + h + ) As h gets closer to 0, the slope gets closer to. The slope is at the point (, ). 0 Cengage Learning. All Rights Reserved. Ma not be scanned, copied or duplicated, or posted to a publicl accessible website, in whole or in part.