Solution Manual for Calculus Single Variable Canadian 8th Edition by Adams Essex

Solution Manual for Calculus Single Variable Canadian
8th Edition by Adams Essex




CHAPTER g;2. g ; g; DIFFERENTIATION Slope g;of g;y g;√
= x g;+ g;1 g;at g;x g;= g;3 g;is

√ √

m g;= g;lim 4 g;+ g;h g;− g;2 g ; · g; √g;4 g ; + g;h g ; + g;2


Section g;2.1 g;Tangent g;Lines g;and g;Their h→0 h 4 g;+ g;h g;+ g;2
Slopes g;(page g;100)
g; 4 g;+ g;h g;− g;4

lim √
h→0
h g;+ g;h g;+ g;2
Slope g;of g;y g;= g;3x g;− g;1 g;at g ; h
1 1
(1, g;2) g;is
g;




h →0 g; g; g; g; g; g; g; g;
√ 4 + h + 2 = 4 g;
.




m g;= g;lim g ; g; 3(1 g;+ g;h g;) g;−1 g;−(3 g;× g;1 g;−1)

lim g ; 3h
g ; = g;3.
h→0 h h→0 g ; g ; h
Tangent g;line g;is g;y g;− g;2 g;= g; g;
1 g;(x g;− g;3), g;or g;x g;− g;4 g;y g;= g;−5.


4
The g;tangent g;line g;is g;y g;− g;2 g;= g;3(x g;− g;1), g;or g;y g;= g;3x 1
g; − g;1. g;(The g;tangent g;to g;a g;straight g;line g;at g;any g;point
g; on g;it g;is g;the g;same g;straight g;line.) at g;x g;= g;9 g;is

Since g;y g;= g;x g;/2 g;is g;a g;straight g;line, g;its g;tangent g;at g;any 8. The g;slope g;of g;y g;=
g; point g;(a, g;√x

m g;= g;lim g ; g1; √ g ; 1 − g;g;1
a/2) g;on g;it g;is g;the g;same g;line g;y g;= g;x g;/2.
g ; g ;
→ h 3 √
h g ; 0 √9 g;+ g;h

3 g;− · g;
3
Slope g;of g;y g;= g;2x g;
2 g;
− g;5 g;at = g;lim √ 9 g ; g ; + g;h

√
g; (2, g;3) g;is g ;
9 g;+ g;h +
g;



h→0 g ; g ; 3h g; g; 9 g; + g;h 3 g;+ 9 g;+ g;h


m g;= g;lim g; g;
2 g;
2(2 g;+ g;h g;) −5 9 g;− g;9 g;− g;h
lim √ √
2 g;
−(2(2 ) g;−5)
g;



h→0 h h→0 g ; 3h g ; g ; 9 g;+ g;h g;(3 g;+ 9 g;+ g;h g;)

, lim g ; 8 g;+ g;8h g;+ g;2h g;2 g;−8 = g;− g ; g; 1 g; g ; = g;− g; g ; 1 g;.


h→0 h 3(3)(6) 54
= g;lim g;(8 g;+ g;2h g; ) g;= g;8
The g;tangent g;line g;at g;(9,3 g ; g ;
1
) 3 is g;y54g;=
g; g; g ;
1 g ;
− g ;
1 g ;
(x
h→0
g;− g;9), g;or
Tangent g;line g;is g;y g;− g;3 g;= g;8(x g;− g;2) g;or g;y g;= y g;= g; 1 g ; − g ; g ; 1 g ; x g;.
2
8x g;− g;13. g;The g;slope g;of g;y g;= g;6 g;− g;x g;− g;x g;
g;
2 54

2x
at g;x g;= g;−2 g;is
g ;


9. Slope g;of g;y g;= x g;+ g;2 g; at g;x 2 g;is

m g;= g;lim6 g;−(−2 g;+ g;h g;) g;−(−2 g;+
g; 2(2 g;+ g;h g;)
2
g;h g; ) g ; −4
− g;1
m g;= g;lim g ; g; 2 g;+ g;h g;+ g ;

h→0 h
2
2
= g;lim3h g;−h g;
g ;
= g;lim g;(3 g;− g;h g;) g;= g;3. h→0 h


h
= g;lim g;4 g;+ g;2h g;−2 g;−h g;−2
h→0 h→0

, SECTIONINSTRUCTOR2.1’S(PAGESOLUTIONS100) MANUAL ADAMS SECTIONandESSEX:2.1CALCULUS(PAGE100)8




The tangent line at (−2, 4) is y = 3x + 10. h→0h (2 + h + 2)
= lim h = 1 .
3
Slope of y = x + 8 at x = −2 is h→0 h (4 + h ) 4
3
m = lim (−2 + h ) + 8 −(−8 + 8) Tangent line is y − 1 = 1 (x − 2),
h→0 h 4

2 3
lim −8 + 12h −6h + h + 8 −0
or x − 4 y = −2. √
2
h→0 h 10. The slope of y = 5−x at x = 1 is

h→0 m = lim p 5 − (1 + h )2 − 2
h→0 h
Tangent line is= yg;lim
2
− 0 g=; 12(x
12 g;−+ + g;yh g;= 12x +
= g;24.
g ;
g;6h 12 2
2)g;or 5 − (1 + h ) − 4
= lim
1 p 2
h→0 h 5 − (1 + h ) + 2
6. The slope of y = x 2 + 1 at (0, 1) is

= lim p −2 − h =− 1
m = lim 1 1 − 1 = lim −h = 0. h→0 5 − (1 + h ) 2+2
2

2 2
h h +1 h +1
2
h→0 h→0 The tangent line at (1, 2) is y = 2 − 1
(x − 1), or




The g;tangent g;line g;at g;(0, g;1) g;is g;y g;= g;1. y g;= g ; 5 g ; − g ; 1 g ; x g;.


2 2

, INSTRUCTORSECTION2.1’S(PAGESOLUTIONS100) MANUAL ADAMS SECTIONandESSEX:2.1CALCULUS(PAGE100)8




2
Slope g;of g;y g;= g;x g; at g;x g;= g;x0 If g;m g;= g;−3, = g;−
g ;
g ; is g ; g ; then g;x g;20 g ;
3
g ; g; g ;. The g;tangent g;line g;with
slope
g;


= g;−3 g;at g;(− 3 g ; , g;5 g ; ) g;is g;y g ; = 5 g ; − g;3(x g;+ 3 g ;


g ; ), g;that g;is,
m
(x g ; +h g ;
2 g;
− 2x g ; h g;+
2 4 4 2



2 2 13
x
g; g;
h
g; g;

)
m g;= g ; g ; lim g ;
g; g; 0 g ; g ; 0 3x g;− g ;
4 g ; g ;
.
= g;lim 0 = g;2x0 g;.



h→0 h h→0 h 3

Tangent g;line g;is g;y g;− g;x g;02 g;= g;2x0 g ; (x g;− g;x0 g ; ), a) g ; Slope g;of g;y g;= g;x at g;x g;= g;a g;is

3 3
or g;y g;= g;2x0 g;x g;− g;x m g;= g ; lim g ; (a g;+ g;h g;) g ; −a g ;


2 g;.
g; 0
h
12. The g;slope g;of g;y g;= 1 g; at
h→0

lim 2 g; 2 g; 3 g; 3
(a,
g; g ; 1 g; g; ) is + g;3a h g;+ g;3ah g; + g;h g; −a

a a3
x g ;




m g;= g;lim g; g; 1 g ; 1 + g;1 = g;lim g ; g ; a g ; −a g ;−h g ; g ; = g;− g; g; 1 g ; .
h→0 h

2 2 g; 2
g ; g ; g ; = g;lim g;(3a + g;3ah g;+ g;h g; ) g;= g;3a

h→0 g ; g ; h g ; a g;+ g;h a h→0

h→0 g ; h g;(a g;+ g;h g;)(a) a
2
2
The g;tangent g;line g;at g;(a, g ; g ; 1 g;) g;is g;y g;= 1 g;− g ;


1 g;
g; g ; g ; (x g;− g;a), g;or
2
a a a

y g;= g ; 2 g;− g ; g; x g;. b) g ; We g;have g;m g;= g;3 g;if g;3a g ; = g;3, g;i.e., g;if g;a g;= g;±1.
3

Lines g;of g;slope g;3 g;tangent g;to g;y = g ; g ;
a2
x
√ are
g; g; g ; g ;
a g ;



y g;= g;1 g;+ g;3(x g;− g;1) g;and g;y g;= g;−1 g;+ g;3(x g;+ g;1), g;or g;y g;= g;3x
g ;

13. Since g;limh→0 |0 g;+ g;h g;| g;− g;0 g ; g ; = g;lim 1 – g;2 g;and g;y g;= g;3x g;+ g;2.
does g;not
h h→0 g ; |h g; |sgn g;(h g;)


exist g;(and g;is g;not g;∞ g;or g;−∞), the g;graph g;of g ; f g;(x |x
The g;slope g;of g;y g;= g;x g;3 g;− g;3x g;at g;x g;= g;a g;is
g;|
√
g;) g;= g;


has g;no g;tangent g;at g;x g;= g;0.
h→0 g ; g ; h
4/3

The g;slope g;of g ; f g;(x g;) g;= g;(x g;− g;1) at g;x g;= g;1
is
g; 1
mlim g;