Multivariable Calculus 9th Edition by James Stewart solution manual

c ° 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.
SECTION 1.4 EXPONENTIALFUNCTIONS
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(c) For any function  with domain , define functions  and  as in parts (a) and (b). Then
1
2  is even,
1
2  is odd, and we
show that () =
1
2 () +
1
2 ():
1
2 () +
1
2 () =
1
2
[() + (−)] +
1
2 [() − (−)]
=
1
2 [() + (−) + () − (−)]
=
1
2 [2()] = ()
as desired.
(d) () = 2  + ( − 3) 2 has domain , so we know from part (c) that () =
1
2  () +
1
2 () where
() = () +  (−) = 2  + ( − 3) 2 + 2 − + (− − 3) 2
= 2  + 2 − + ( − 3) 2 + ( + 3) 2
and () = () −  (−) = 2  + ( − 3) 2 − [2 − + (− − 3) 2 ]
= 2  − 2 − + ( − 3) 2 − ( + 3) 2
1.4 Exponential Functions
1. (a)
−2 6
4 3
=
−2 6
(2 2 ) 3
= − 2
6
2 6
= −1 (b)
(−3) 6
9 6
=
 −3
9
 6
=

− 1
3
 6
=
1
3 6
(c)
1
4
√ 
5
=
1
4
√ 
4 · 
=
1

4
√  (d)
 3 ·  
 +1
=
 3+
 +1
=  (3+)−(+1) =  2
(e)  3 (3 −1 ) −2 =  3 3 −2 ( −1 ) −2 =
 3 ·  2
3 2
=
 5
9
(f)
2 2 
(3 −2 ) 2
=
2 2 
3 2 ( −2 ) 2  2
=
2 2 
9 −4  2
=
2
9 
2−(−4)  1−2
=
2
9 
6  −1
=
2 6
9
2. (a)
3
√ 4
3
√ 108 =
3
√ 4
3
√ 4 · 27 =
3
√ 4
3
√ 4 ·
3
√ 27 =
1
3
√ 27 =
1
3
(b) 27 23 = (27 13 ) 2 =

3
√ 27  2
= 3 2 = 9
(c) 2 2 (3 5 ) 2 = 2 2 · 3 2 ( 5 ) 2 = 2 2 · 9 10 = 2 · 9 2+10 = 18 12
(d) (2 −2 ) −3  −3 = 2 −3 ( −2 ) −3  −3 =
 6 ·  −3
2 3
=
 6+(−3)
8
=
 3
8
(e)
3 32 ·  12
 −1
= 3 32+12 ·  1 = 3 2 ·  = 3 3
(f)

 √ 
3
√  =
( 12 ) 12
() 13
=
 12 ( 12 ) 12
 13  13
=
 12  14
 13  13
=  12−13  14−13 =  16  −112 =
 16
 112
=
6
√ 
12
√ 
3. (a) () =   ,   0 (b)  (c) (0∞) (d) See Figures 4(c), 4(b), and 4(a), respectively.
4. (a) The number  is the value of  such that the slope of the tangent line at  = 0 on the graph of  =   is exactly 1.
(b)  ≈ 271828 (c) () =  
c ° 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.
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CHAPTER 1 FUNCTIONSAND MODELS
5. All of these graphs approach 0 as  → −∞, all of them pass through the point
(01), and all of them are increasing and approach ∞ as  → ∞. The larger the
base, the faster the function increases for   0, and the faster it approaches 0 as
 → −∞.
Note: The notation “ → ∞” can be thought of as “ becomes large” at this point.
More details on this notation are given in Chapter 2.
6. The graph of  − is the reflection of the graph of   about the ­axis, and the
graph of 8 − is the reflection of that of 8  about the ­axis. The graph of 8 
increases more quickly than that of   for   0, and approaches 0 faster
as  → −∞.
7. The functions with base greater than 1 (3  and 10  ) are increasing, while those