TABLE OF CONTENTS1 Functions 1 1.1 Functions and Their Graphs 1 1.2 Combining Functions; Shifting and Scaling Graphs 8 1.3 Trigonometric Functions 19 1.4 Graphing with Calculators and Co
Trang 3The author and publisher of this book have used their best efforts in preparing this book These effortsinclude the development, research, and testing of the theories and programs to determine theireffectiveness The author and publisher make no warranty of any kind, expressed or implied, with regard
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Trang 4This Instructor's Solutions Manual contains the solutions to every exercise in the 12th Edition of THOMAS' CALCULUS
by Maurice Weir and Joel Hass, including the Computer Algebra System (CAS) exercises The corresponding Student'sSolutions Manual omits the solutions to the even-numbered exercises as well as the solutions to the CAS exercises (becausethe CAS command templates would give them all away)
In addition to including the solutions to all of the new exercises in this edition of Thomas, we have carefully revised orrewritten every solution which appeared in previous solutions manuals to ensure that each solution
ì conforms exactly to the methods, procedures and steps presented in the text
ì is mathematically correct
ì includes all of the steps necessary so a typical calculus student can follow the logical argument and algebra
ì includes a graph or figure whenever called for by the exercise, or if needed to help with the explanation
ì is formatted in an appropriate style to aid in its understanding
Every CAS exercise is solved in both the MAPLE and MATHEMATICA computer algebra systems A template showing
an example of the CAS commands needed to execute the solution is provided for each exercise type Similar exercises withinthe text grouping require a change only in the input function or other numerical input parameters associated with the problem(such as the interval endpoints or the number of iterations)
For more information about other resources available with Thomas' Calculus, visit http://pearsonhighered.com
Trang 5TABLE OF CONTENTS
1 Functions 1
1.1 Functions and Their Graphs 1
1.2 Combining Functions; Shifting and Scaling Graphs 8
1.3 Trigonometric Functions 19
1.4 Graphing with Calculators and Computers 26
Practice Exercises 30
Additional and Advanced Exercises 38
2 Limits and Continuity 43
2.1 Rates of Change and Tangents to Curves 43
2.2 Limit of a Function and Limit Laws 46
2.3 The Precise Definition of a Limit 55
3.1 Tangents and the Derivative at a Point 93
3.2 The Derivative as a Function 99
3.3 Differentiation Rules 109
3.4 The Derivative as a Rate of Change 114
3.5 Derivatives of Trigonometric Functions 120
3.6 The Chain Rule 127
4.1 Extreme Values of Functions 167
4.2 The Mean Value Theorem 179
4.3 Monotonic Functions and the First Derivative Test 188
4.4 Concavity and Curve Sketching 196
5.1 Area and Estimating with Finite Sums 257
5.2 Sigma Notation and Limits of Finite Sums 262
5.3 The Definite Integral 268
5.4 The Fundamental Theorem of Calculus 282
5.5 Indefinite Integrals and the Substitution Rule 290
5.6 Substitution and Area Between Curves 296
Practice Exercises 310
Additional and Advanced Exercises 320
Trang 66.1 Volumes Using Cross-Sections 327
6.2 Volumes Using Cylindrical Shells 337
6.3 Arc Lengths 347
6.4 Areas of Surfaces of Revolution 353
6.5 Work and Fluid Forces 358
6.6 Moments and Centers of Mass 365
Additional and Advanced Exercises 528
9 First-Order Differential Equations 537
9.1 Solutions, Slope Fields and Euler's Method 537
9.2 First-Order Linear Equations 543
9.3 Applications 546
9.4 Graphical Solutions of Autonomous Equations 549
9.5 Systems of Equations and Phase Planes 557
Practice Exercises 562
Additional and Advanced Exercises 567
10 Infinite Sequences and Series 569
10.1 Sequences 569
10.2 Infinite Series 577
10.3 The Integral Test 583
10.4 Comparison Tests 590
10.5 The Ratio and Root Tests 597
10.6 Alternating Series, Absolute and Conditional Convergence 60210.7 Power Series 608
10.8 Taylor and Maclaurin Series 617
10.9 Convergence of Taylor Series 621
10.10 The Binomial Series and Applications of Taylor Series 627Practice Exercises 634
Additional and Advanced Exercises 642
Trang 710.5 The Ratio and Root Tests 597
10.6 Alternating Series, Absolute and Conditional Convergence 602
10.7 Power Series 608
10.8 Taylor and Maclaurin Series 617
10.9 Convergence of Taylor Series 621
10.10 The Binomial Series and Applications of Taylor Series 627
Practice Exercises 634
Additional and Advanced Exercises 642
11 Parametric Equations and Polar Coordinates 647
11.1 Parametrizations of Plane Curves 647
11.2 Calculus with Parametric Curves 654
11.3 Polar Coordinates 662
11.4 Graphing in Polar Coordinates 667
11.5 Areas and Lengths in Polar Coordinates 674
11.6 Conic Sections 679
11.7 Conics in Polar Coordinates 689
Practice Exercises 699
Additional and Advanced Exercises 709
12 Vectors and the Geometry of Space 715
12.1 Three-Dimensional Coordinate Systems 715
12.2 Vectors 718
12.3 The Dot Product 723
12.4 The Cross Product 728
12.5 Lines and Planes in Space 734
12.6 Cylinders and Quadric Surfaces 741
Practice Exercises 746
Additional Exercises 754
13 Vector-Valued Functions and Motion in Space 759
13.1 Curves in Space and Their Tangents 759
13.2 Integrals of Vector Functions; Projectile Motion 764
13.3 Arc Length in Space 770
13.4 Curvature and Normal Vectors of a Curve 773
13.5 Tangential and Normal Components of Acceleration 778
13.6 Velocity and Acceleration in Polar Coordinates 784
Practice Exercises 785
Additional Exercises 791
Trang 814.1 Functions of Several Variables 795
14.2 Limits and Continuity in Higher Dimensions 804
14.3 Partial Derivatives 810
14.4 The Chain Rule 816
14.5 Directional Derivatives and Gradient Vectors 824
14.6 Tangent Planes and Differentials 829
14.7 Extreme Values and Saddle Points 836
14.8 Lagrange Multipliers 849
14.9 Taylor's Formula for Two Variables 857
14.10 Partial Derivatives with Constrained Variables 859
Practice Exercises 862
Additional Exercises 876
15 Multiple Integrals 881
15.1 Double and Iterated Integrals over Rectangles 881
15.2 Double Integrals over General Regions 882
15.3 Area by Double Integration 896
15.4 Double Integrals in Polar Form 900
15.5 Triple Integrals in Rectangular Coordinates 904
15.6 Moments and Centers of Mass 909
15.7 Triple Integrals in Cylindrical and Spherical Coordinates 914
15.8 Substitutions in Multiple Integrals 922
Practice Exercises 927
Additional Exercises 933
16 Integration in Vector Fields 939
16.1 Line Integrals 939
16.2 Vector Fields and Line Integrals; Work, Circulation, and Flux 944
16.3 Path Independence, Potential Functions, and Conservative Fields 952
16.4 Green's Theorem in the Plane 957
16.5 Surfaces and Area 963
Trang 9CHAPTER 1 FUNCTIONS
1.1 FUNCTIONS AND THEIR GRAPHS
1 domainœ _ß _( ); rangeœ[1ß _) 2 domainœ[0ß _); rangeœ _ß( 1]
3 domainœ Ò ß _2 ); y in range and yœ È5x10 ! Êy can be any positive real numberÊrangeœ Ò!ß _)
4 domainœ _ß Ó Ò _( 0 3, ); y in range and yœÈx23x ! Êy can be any positive real numberÊrangeœ Ò!ß _)
5 domainœ _ß Ñ Ð _( 3 3, ); y in range and yœ 34t, now if t3 Ê3 ! Êt 34t !, or if t3
nonzero real numberÊrangeœ Ð_ß Ó Ð!ß _18 )
7 (a) Not the graph of a function of x since it fails the vertical line test
(b) Is the graph of a function of x since any vertical line intersects the graph at most once
8 (a) Not the graph of a function of x since it fails the vertical line test
(b) Not the graph of a function of x since it fails the vertical line test
9 baseœx; (height)#ˆ ‰#x #œx # Ê heightœ #3x; area is a(x)œ #"(base)(height)œ #"(x)Š #3x‹œ 3x ;#
4
perimeter is p(x)œ œx x x 3x
10 sœside length Ê s#s#œd # Ê sœ Èd2; and area is aœs # Ê aœ "#d#
11 Let Dœdiagonal length of a face of the cube and j œthe length of an edge Then j # D#œd and#
D#œ j Ê j œ2 # 3 # d # Ê j œ d The surface area is 6j œ# 6d œ2d and the volume is # j œ$ d $Î#œ d
Trang 1015 The domain is a_ß _b 16 The domain is a_ß _b.
17 The domain is a_ß _b 18 The domain is Ð_ß !Ó
19 The domain is a_ß ! !ß _b a b 20 The domain is a_ß ! !ß _b a b
21 The domain is a_ß 5b Ð ß Ó Ò5 3 3, 5Ñ a5, _b 22 The range is 2, 3 Ò Ñ
23 Neither graph passes the vertical line test
Trang 11Section 1.1 Functions and Their Graphs 3
24 Neither graph passes the vertical line test
30 (a) Line through a!ß2 and b a#ß !b: yœ x 2
Line through 2a ß "b and a&ß !b: mœ ! "& # œ "$ œ "$, so yœ "$ax2b " œ "$x&$
(b) Line through a"ß !b and a!ß $b: mœ! Ð"Ñ$ ! œ $, so yœ $ $x
Line through a!ß $b and a#ß "b: mœ " $# ! œ %# œ #, so yœ # $x
œ $ $ " Ÿ !
# $ ! Ÿ #œ
Trang 1231 (a) Line through a"ß "b and a!ß !b: yœ x
Line through a!ß "b and a"ß "b: yœ "
Line through a"ß "b and a$ß !b: mœ !"$" œ"# œ "#, so yœ "#ax " " œ b "#x$#
Line through 1a ß 1 and 3b a ß 1 : yb œ 1
Ÿ Ÿ
1 2
32 (a) Line through ˆT ‰ and Ta b: m , so y ˆx T‰ 0 x
T T
33 (a) xÚ Û œ0 for x−[0 1) ß (b) xÜ Ý œ0 for x− ß( 1 0]
34 xÚ Û œ Ü Ýx only when x is an integer
35 For any real number x, nŸ Ÿ "x n , where n is an integer Now: nŸ Ÿ " Ê Ð "Ñ Ÿ Ÿ x n n x n By definition: Ü Ý œ x n and xÚ Û œ Ê Ú Û œ n x n So Ü Ý œ Ú Ûx x for all x− d
36 To find f(x) you delete the decimal or
fractional portion of x, leaving only
the integer part
Trang 13Section 1.1 Functions and Their Graphs 5
37 Symmetric about the origin 38 Symmetric about the y-axis
39 Symmetric about the origin 40 Symmetric about the y-axis
Trang 1443 Symmetric about the origin 44 No symmetry
47 Since a horizontal line not through the origin is symmetric with respect to the y-axis, but not with respect to the origin, thefunction is even
48 f xa bœx& œ " and fa bx œ a bx &œ " œ ˆ ‰" œ f x Thus the function is odd.a b
49 Since f xa bœx# " œ a bx # " œ f x The function is even.a b
50 Since f xÒa bœx# Ó Á Ò x fa bx œ a bx # Óx and f xÒa bœx# Ó Á Òx f xa bœ a bx # Óx the function is neither even nor odd
51 Since g xa bœx$x, ga bx œ x$ œ x ax$xbœ g x So the function is odd.a b
52 g xa bœx% $x# " œ a bx % $ a bx # " œga b ßx thus the function is even
53 g xa bœx #" " œ a bx"#" œga bx Thus the function is even
54 g xa bœ x ; ga bx œ x œ g x So the function is odd.a b
x # " x#"
55 h ta bœ t "" ; ha b œt "t " ; h t a bœ" "t Since h ta bÁ h t and h ta b a bÁha bt , the function is neither even nor odd
56 Since t |l $ œ l a bt $|, h ta bœha bt and the function is even
Trang 15Section 1.1 Functions and Their Graphs 7
57 h ta bœ2t ", ha b œ "t 2t So h ta bÁha bt h t a bœ "2t , so h ta bÁ h t The function is neither even nora b odd
58 h ta bœ l l "2 t and ha b œ l l " œ l l "t 2 t 2 t So h ta bœha bt and the function is even
64 (a) Let hœheight of the triangle Since the triangle is isosceles, AB# AB#œ2#ÊAB œÈ2 So,Þ
h# " œ# ŠÈ2‹#Êh œ " ÊB is at a!ß " Êb slope of ABœ " ÊThe equation of AB is
yœf(x)œ "x ; x− Ò!ß "Ó
(b) A xÐ Ñ œ2x yœ2xÐ "Ñ œ x 2x# #x; x− Ò!ß "Ó
65 (a) Graph h because it is an even function and rises less rapidly than does Graph g
(b) Graph f because it is an odd function
(c) Graph g because it is an even function and rises more rapidly than does Graph h
66 (a) Graph f because it is linear
(b) Graph g because it contains a!ß "b
(c) Graph h because it is a nonlinear odd function
67 (a) From the graph, x# 1 4x Ê x− ß( 2 0) %ß _( )
Solution interval: (#ß0) %ß _( )
Trang 1668 (a) From the graph, x 13 x 12 Ê x− _ß ß( 5) ( 1 1)
72 (a) Note that 2 mi = 10,560 ft, so there are È800#x feet of river cable at $180 per foot and 10,560# a x feet of landb
cable at $100 per foot The cost is C xa bœ180È800#x#100 10,560a x b
Trang 17Section 1.2 Combining Functions; Shifting and Scaling Graphs 9
(d) Since f g xa‰ ba bœfˆÈx‰œ l lx , f xa bœx (Note that the domain of the composite is # Ò!ß _Ñ.)
The completed table is shown Note that the absolute value sign in part (d) is optional
16 (a) f g 0a a b œ œ œb fa b1 2 a b1 3, where g 0a bœ œ 0 1 1
(b) g f 3aa b œ œ œb ga b1 a b1 1, where f 3a bœ œ 2 3 1
(c) g ga a b1 bœg 1a bœ œ1 1 0, where ga b1 œ a b1 œ1
Trang 19Section 1.2 Combining Functions; Shifting and Scaling Graphs 11
Trang 21Section 1.2 Combining Functions; Shifting and Scaling Graphs 13
53 54
55 (a) domain: [0 2]; range: [ß #ß $] (b) domain: [0 2]; range: [ 1 0]ß ß
(c) domain: [0 2]; range: [0 2] ß ß (d) domain: [0 2]; range: [ 1 0]ß ß
(e) domain: [ 2 0]; range: [ ß !ß1] (f) domain: [1 3]; range: [ß !ß "]
(g) domain: [ 2 0]; range: [ ß !ß "] (h) domain: [ 1 1]; range: [ ß !ß "]
Trang 2256 (a) domain: [0 4]; range: [ 3 0] ß ß (b) domain: [ 4 0]; range: [ ß !ß $]
(c) domain: [ 4 0]; range: [ ß !ß $] (d) domain: [ 4 0]; range: [ ß "ß %]
(e) domain: [#ß4]; range: [ 3 0] ß (f) domain: [ 2 2]; range: [ 3 0] ß ß
(g) domain: ["ß5]; range: [ 3 0] ß (h) domain: [0 4]; range: [0 3]ß ß
Trang 23Section 1.2 Combining Functions; Shifting and Scaling Graphs 15
67 Let yœ # " œÈ x f x and let g xa b a bœx"Î#,
vertically by a factor of È#; and the graph of
j xa bœf x is the graph of i x reflected acrossa b a b
f x The graph of g x is the
œ " œÈ x a bÞ a b
#
graph of yœ Èx reflected across the x-axis
The graph of h x is the graph of g x shifteda b a b
right two units And the graph of i x is thea b
graph of h x compressed vertically by a factora b
of È#
69 yœf xa bœx Shift f x one unit right followed by a$ a b
shift two units up to get g xa bœax " #b3
70 yœ " B # œ Ò " # Ó œa b$ ax b$ a b f x a b
Let g xa bœx , h x$ a bœax "b a b$, i x œax " #b$ a b,
and j xa bœ Ò " # Óax b$ a b The graph of h x is thea b
graph of g x shifted right one unit; the graph of i x isa b a b
the graph of h x shifted down two units; and the grapha b
of f x is the graph of i x reflected across the x-axis.a b a b
71 Compress the graph of f xa b œ " horizontally by a factor
x
of 2 to get g xa bœ " Then shift g x vertically down 1a b
#xunit to get h xa b œ " "
#x
Trang 25Section 1.2 Combining Functions; Shifting and Scaling Graphs 17
major axis, AB, is the segment from a)ß $b a to !ß $b
84 The ellipse x%#y#&# œ " has center h, ka bœ !ß !a b
Shifting the ellipse 3 units right and 2 units down
produces an ellipse with center at h, ka bœ $ß #a b
and an equation ax %3b# y ##&a b‘ Center,
#
C, is 3a ß #b, and AB, the segment from a$ß $b to
is the major axis
Trang 27Section 1.3 Trigonometric Functions 19
3
2 3
È
È
È
ÈÈ
sin cos
9 sin xœ È83 , tan xœ È8 10 sin xœ1312, tan xœ 125
11 sin xœ " , cos xœ 12 cos xœ , tan xœ "
Trang 29Section 1.3 Trigonometric Functions 21
27 (a) Cos x and sec x are positive for x in the interval
, ; and cos x and sec x are negative for x in the
ˆ1 1‰
2 2
intervals ˆ3 , ‰ and ˆ , 3 ‰ Sec x is undefined
2 1 2 1 2 1 2 1when cos x is 0 The range of sec x is
(_ß "ß _1] [ ); the range of cos x is ["ß1]
(b) Sin x and csc x are positive for x in the intervals
, and , ; and sin x and csc x are negative
ˆ3 ‰ a! b
for x in the intervals a !1, and b ˆ1, 3 ‰ Csc x is
2 1undefined when sin x is 0 The range of csc x is
(_ß ß _1] [1 ); the range of sin x is ["ß "]
28 Since cot xœ " , cot x is undefined when tan xœ0
tan xand is zero when tan x is undefined As tan x approaches
zero through positive values, cot x approaches infinity
Also, cot x approaches negative infinity as tan x
approaches zero through negative values
35 cos (AB) œcos (A ( B))œcos A cos ( B) sin A sin ( B) œcos A cos Bsin A ( sin B)
cos A cos B sin A sin B
36 sin (AB) œsin (A ( B))œsin A cos ( B) cos A sin ( B) œsin A cos Bcos A ( sin B)
sin A cos B cos A sin B
37 If BœA, A œB 0 Ê cos (AB)œcos 0œ1 Also cos (AB)œcos (AA)œcos A cos Asin A sin Acos A sin A Therefore, cos A sin A 1
Trang 3038 If Bœ2 , then cos (A1 2 )1 œcos A cos 21sin A sin 21œ(cos A)(1)(sin A)(0)œcos A and
sin (A2 )1 œsin A cos 21cos A sin 21œ(sin A)(1)(cos A)(0)œsin A The result agrees with the
fact that the cosine and sine functions have period 2 1
39 cos (1x)œcos cos 1 B sin sin x1 œ ( 1)(cos x)(0)(sin x)œ cos x
40 sin (21x)œsin 2 cos ( x)1 cos (2 ) sin ( x)1 œ(0)(cos ( x)) (1)(sin ( x)) œ sin x
41 sinˆ3 1 x‰ sinˆ ‰3 1 cos ( x) cosˆ ‰3 1 sin ( x) ( 1)(cos x) (0)(sin ( x)) cos x
54 cos 2)cos)œ Ê0 2cos2) 1 cos)œ Ê0 2cos2)cos) œ Ê1 0 acos)1 2cosba )1bœ0
55 tan (AB)œ sin (A B) œ œ
cos (A B) cos A cos B sin A sin B
sin A cos B cos A cos B
sin A cos B cos A sin B cos A cos B cos A cos B cos A cos B sin A sin B cos A cos B cos A cos B œ tan A tan B
1 tan A tan B
56 tan (AB)œ sin (A B) œ œ
cos (A B) cos A cos B sin A sin B
sin A cos B cos A cos B
sin A cos B cos A sin B cos A cos B cos A cos B cos A cos B sin A sin B cos A cos B cos A cos B œ tan A tan B
1 tan A tan B
57 According to the figure in the text, we have the following: By the law of cosines, c#œa#b#2ab cos )
1 1 2 cos (A B) 2 2 cos (A B) By distance formula, c (cos A cos B) (sin A sin B)
Trang 31Section 1.3 Trigonometric Functions 23
58 (a) cos Aa Bbœcos A cos B sin A sin B
sin )œcosˆ1)‰ and cos )œsinˆ1)‰
(b) cos Aa Bbœcos A cos B sin A sin B
cos Aa a Bbbœcos A cos aB b sin A sin aBb
cos A B cos A cos B sin A sin B cos A cos B sin A sin B
cos A cos B sin A sin B
Because the cosine function is even and the sine functions is odd
59 c#œa#b#2ab cos Cœ2#3#2(2)(3) cos (60°)œ 4 9 12 cos (60°)œ1312ˆ ‰"# œ7
Thus, cœÈ7¸2.65
60 c#œa#b#2ab cos Cœ2#3#2(2)(3) cos (40°)œ1312 cos (40°) Thus, cœÈ1312 cos 40°¸1.951
61 From the figures in the text, we see that sin Bœ h If C is an acute angle, then sin Cœh On the other hand,
if C is obtuse (as in the figure on the right), then sin Cœsin (1C)œ h Thus, in either case,
b
hœb sin Cœc sin B Ê ahœab sin Cœac sin B
By the law of cosines, cos Cœa b c and cos Bœa c b Moreover, since the sum of the
(b) In degree mode, when x is near zero degrees the sine of x is much closer to zero than x itself The
curves look like intersecting straight lines near the origin when the calculator is in degree mode
Trang 32f_list := [seq( f(x), B=[1,3,2*Pi,5*Pi] )];
plot( f_list, x=-4*Pi 4*Pi, scaling=constrained,
Trang 33Section 1.3 Trigonometric Functions 25
69 (a) The graph stretches horizontally
(b) The period remains the same: periodœ l lB The graph has a horizontal shift of period."
#
70 (a) The graph is shifted right C units
(b) The graph is shifted left C units
(c) A shift of „one period will produce no apparent shift C l l œ '
71 (a) The graph shifts upwards D units for Dl l !
(b) The graph shifts down D units for Dl l !Þ
72 (a) The graph stretches A units l l (b) For A !, the graph is inverted
Trang 341.4 GRAPHING WITH CALCULATORS AND COMPUTERS
1-4 The most appropriate viewing window displays the maxima, minima, intercepts, and end behavior of the graphs and has little unused space
5-30 For any display there are many appropriate display widows The graphs given as answers in Exercises 5 30
are not unique in appearance
5 Ò ß Ó2 5 by Ò ß15 40 Ó 6 Ò ß Ó4 4 by Ò ß Ó4 4
7 Ò ß Ó2 6 by Ò250 50 ß Ó 8 Ò ß Ó1 5 by Ò ß5 30Ó
Trang 35Section 1.4 Graphing with Calculators and Computers 27
9 4 4 by 5 5 Ò ß Ó Ò ß Ó 10 2 2 by 2 8Ò ß Ó Ò ß Ó
11 2 6 by 5 4 Ò ß Ó Ò ß Ó 12 4 4 by 8 8Ò ß Ó Ò ß Ó
13 1 6 by 1 4 Ò ß Ó Ò ß Ó 14 1 6 by 1 5Ò ß Ó Ò ß Ó
15 3 3 by Ò ß Ó Ò ß Ó0 10 16 1 2 by Ò ß Ó Ò ß Ó0 1
Trang 3617 5 1 by 5 5 Ò ß Ó Ò ß Ó 18 5 1 by 2 4Ò ß Ó Ò ß Ó
19 4 4 by Ò ß Ó Ò ß Ó0 3 20 5 5 by 2 2Ò ß Ó Ò ß Ó
21 by Ò"!ß "!Ó Ò'ß 'Ó 22 Ò&ß &Óby Ò#ß #Ó
23 by Ò'ß "!Ó Ò'ß 'Ó 24 Ò$ß &Óby Ò#ß "!Ó
Trang 37Section 1.4 Graphing with Calculators and Computers 29
Trang 3833 34
CHAPTER 1 PRACTICE EXERCISES
1 The area is Aœ1r and the circumference is C# œ #1r Thus, rœ Ê œA 1 œ
Trang 39Chapter 1 Practice Exercises 31
3 The coordinates of a point on the parabola are x x The angle of inclination joining this point to the origin satisfiesa ß #b )the equation tan )œx œx Thus the point has coordinates x xß œ tan )ßtan )
10 ya œ xb a xb& a xb$ œ a xb x& x$ œ x y x Odd.a b
11 ya œ " xb cosa œ " xb cos xœy x Even.a b
12 ya œxb secax tanb a œxb sin x œ œ sec x tan xœ y x Odd.a b
14 ya œ xb a xb sina œ xb a xb sin xœ ax sin xbœ y x Odd.a b
15 ya œ xb x cosa œ xb x cos x Neither even nor odd
16 ya œ xb a x cosb a œ xb x cos xœ y x Odd.a b
17 Since f and g are odd Ê œ fa xb f x and ga b a œ xb g x a b
(a) af g† ba œ xb fa x gb a œ Òxb f xa bÓÒg xa bÓ œf x g xa b a bœ †af g xba bÊ †f g is even
(b) f3a œ xb fa x fb ax fb a œ Òxb f xa bÓÒf xa bÓÒf xa bÓ œ f xa b a b a b†f x †f x œ f x3a bÊf is odd.3
(c) f sina axbbœ fa sin xa bbœ f sin xa a bbÊf sin x is odd.a a bb
(d) g seca axbbœg sec xa a bbÊg sec x is even.a a bb
(e) gl l œ la xb g xa bl œ lg xa bl Ê l lg is evenÞ
Trang 4018 Let f aa xbœf aa x and define g xb a bœf xa a Then gb a œxb faa xb abœf aa xbœf aa xbœf xa abœg xa b
g x f x a is even
Ê a bœ a b
19 (a) The function is defined for all values of x, so the domain is a_ß _b
(b) Since x attains all nonnegative values, the range is l l Ò#ß _Ñ
20 (a) Since the square root requires " !x , the domain is Ð_ß "Ó
(b) Since È" x attains all nonnegative values, the range is Ò#ß _Ñ
21 (a) Since the square root requires "' x# !, the domain is Ò%ß %Ó
(b) For values of x in the domain, ! Ÿ "' x#Ÿ "', so ! Ÿ "' È x#Ÿ % The range is Ò!ß %Ó
22 (a) The function is defined for all values of x, so the domain is a_ß _b
(b) Since $#x attains all positive values, the range is a"ß _b
23 (a) The function is defined for all values of x, so the domain is a_ß _b
(b) Since e# x attains all positive values, the range is a$ß _b
24 (a) The function is equivalent to yœtan x, so we require x# # Á k 1 for odd integers k The domain is given by xÁ k 1for
odd integers k
(b) Since the tangent function attains all values, the range is a_ß _b
25 (a) The function is defined for all values of x, so the domain is a_ß _b
(b) The sine function attains values from " " to , so # Ÿ #sin xa$ 1bŸ # and hence $ Ÿ #sin xa$ 1b " Ÿ " The range is Ò ß Ó3 1
26 (a) The function is defined for all values of x, so the domain is a_ß _b
(b) The function is equivalent to yœÈ&x , which attains all nonnegative values The range is # Ò!ß _Ñ
27 (a) The logarithm requires x $ !, so the domain is a$ß _b
(b) The logarithm attains all real values, so the range is a_ß _b
28 (a) The function is defined for all values of x, so the domain is a_ß _b
(b) The cube root attains all real values, so the range is a_ß _b
29 (a) Increasing because volume increases as radius increases
(b) Neither, since the greatest integer function is composed of horizontal (constant) line segments
(c) Decreasing because as the height increases, the atmospheric pressure decreases
(d) Increasing because the kinetic (motion) energy increases as the particles velocity increases
30 (a) Increasing on 2, Ò _Ñ (b) Increasing on Ò _Ñ1,
(c) Increasing on a_ _, b (d) Increasing on , Ò _Ñ"
#
31 (a) The function is defined for % Ÿ Ÿ %x , so the domain is Ò%ß %Ó
(b) The function is equivalent to yœ l l % Ÿ Ÿ %È x , x , which attains values from to for x in the domain The! # range is Ò!ß #Ó