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(Answers to Exercises for Chapter 7: Logarithmic and Exponential Functions) A.7.1 CHAPTER 7: LOGARITHMIC and EXPONENTIAL FUNCTIONS SECTION 7.1: INVERSE FUNCTIONS 1) a) 3 b) f ± 1 x () = x ± 4 3 , or 1 3 x ± 4 3 c) 1 3 , which is the reciprocal of 3 2) a) 12 b) gx = x 3 , or x 1/3 c) 1 12 , which is the reciprocal of 12

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(Answers to Exercises for Chapter 7: Logarithmic and Exponential Functions) A.7.2 SECTION 7.2: ln x 1) a) 15 x 2 ± 1 5 x 3 ± x + 1 b) 3 x + 2 x 2 + x , or 3 x + 2 xx + 1 () . (Remember to simplify!) c) 3 3 x + 7 d) ± 40 7 ± 4 t , or 40 4 t ± 7 . Hint: Use the Power Rule of Logs first. e) 3 + 3ln x 2 x , or 31 + ln x 2 ± ² ³ ´ µ x . Hint: Use the Power Rule of Logs on the first term. f) ± 2 w . Hint: ln 1 w ± ² ³ ´ µ = ln w · 1 = · ln w . g) w 1 + 2ln w + w 2 ± ² ³ ´ µ 1 + ln w 2 2) 12 x 3 x 4 + 1 + 1 2 x ± 15 3 x ± 4 , or 12 x 3 x 4 + 1 + 1 2 x + 15 4 ± 3 x . As a single fraction: 45 x 5 ± 100 x 4 ± 27 x ± 4 2 4 + 1 3 x ± 4 . Hint: ln x 4 + 1 3 x 3 x ± 4 5 ² ³ ´ ´ ´ µ · · · = x 4 + 1 + 1 2 ln x ± 5ln 3 x ± 4 after applying the laws of logarithms.
(Answers to Exercises for Chapter 7: Logarithmic and Exponential Functions) A.7.3 3) tan x . We then (finally) have: tan xdx ± = ln sec x + C . 4) a) and b) ± 7sin ² cos 6 . Note: You may have obtained ± 7tan cos 7 for part a). This is equivalent to ± cos 6 , where cos ± ² 0 . Logarithmic Differentiation does not apply for values of that make cos = 0 in this problem. 5) a) 351 x 2 + 80 x + 2 () 3 x 2 + 2 3 23 x + 5 , or (if you rationalize the denominator) x 2 + 80 x + 2 3 x 2 + 2 3 3 x + 5 x + 5 b) 24 x 3 x 2 + 2 + 3 x + 5 ± ² ³ ³ ´ µ 3 x 2 + 2 4 3 x + 5 c) Your answer should effectively be the same as your answer to part a). 6) a) 1, ± . Hint: We require x > 0 and ln x > 0 . b) 1 x ln x c) ± 1 + ln x x 2 ln x 2 , or ± 1 + ln x x ln x 2 d) On the x -interval ± , which is the domain of f , ± fx > 0 and ±± < 0 . Therefore, f is increasing and the graph of f is concave down on the x -interval ± .

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