1. Yes. Vertex: $ (0,0) $ , Focus: $ (0,1/2) $ , Direcrtix: y = -1/2

  2. $ y = x^{2} $

  3. $ y = x^{2} $

  4. V: $ (2,1) $ , F: $ (2,) $ , Directrix: $ y= $

  5. V: $ (-1,4) $ , F: $ (-1,) $ , Directrix: $ y= $

Chapter 7

  1. $ f(1.5) = 2 $ 4. After 1 minute, there are 164 bacteria. After an hour, over 777 trillion bacteria.

    1. 13 times (unless you are over 6’8” or under 3’4”) b) 41 times

  3. $2,040; $2,081; $2,972; $104,970

  4. Check your graphs with a graphing program. Here are the intercepts and asymptotes:

\[ \textcircled{d}\left(0,1\right),y=0\quad\textcircled{b}\left(0,42\right),y=0\quad\textcircled{c}\left(0,20\right),y=0\quad\textcircled{d}\left(0,8\right),y=5 \]

  1. $ (0,2) $ , y = -10 f) $ (0,1) $ , y = 0

  1. In 2050, 10.6 billion. In 2075, 13.6 billion. In 2100, 17.4 billion

  2. $ A(t) = 1050 + 10t $ , where t is the number of years since 2000.

Thus, the population in 2050 is projected to be $ A(50) = 1550 $ .

  1. $ A(t) = 2464(1.1)^{t} $ . In 2050, the population is projected to be $ A(50) ,251 $ .

    1. 1 b) 4 c) 216 d) 1/64 e) 1

  3. The net effect is that the price of admission is reduced by 1%. 13. $ M(t) = 1000(.981)^{t} $

  4. No. Compare Exercise 11. In fact, the growth rate here will be about 1.937% per year.

  5. (Check your graphs with a graphing program.) Here are the intercepts and asymptotes:

  1. $ (0,1) $ , y = 0

  2. $ (0,232) $ , y = 0

  3. $ (0,) $ , y = 0

  4. $ (0,8) $ , y = 5

  5. $ (0,-22) $ , y = -10

  6. $ (0,1) $ , y = 0

  7. $ (0,1/2) $ , y = 1

    1. $ y = (1/5)^{t} $ . The function loses 80% of its value in each unit of time.
  1. $ y = 2(1/2)^{t} $ . 50%. c) $ y = (5/2)(3/4)^{t} $ . 25%.
    1. $ y = (100/61)^{-t} $ . 39%. b) $ y = 4(10)^{-t} $ . 90%. c) $ y = (7/8)(5/3)^{-t} $ . 40%.

19. a) 32 b) 5x c) 1 d) 5 e) $ x^{2} + 1 $ f) 1

    1. $ g^{-1}(8) = 2 $ b) $ h^{-1}(-2) = -1 $ c) $ k^{-1}(2) = $ e) $ g^{-1}(10) = - $ , $ [g(10)]^{-1} = 11 $

  2. c, e, and h are false.

    1. $ $ , b) $ $ , c) $ $ , d) $ ( + 5) $

\[ \left)\frac{\ln5}{\ln1.03}\approx54.449,\ f)\approx58.686g\right)\approx13.412,\ h)\approx0.868,\ i)\approx-0.058,\ j)\approx-25.543 \]

    1. $ (2x^{7}) $ b) $ (28) $ c) $ (2x) $ 22. a) $ 2a + 3b - 5c $ b) $ x + y $ c) $ 4y $
  2. $ e^{10} $ , $ e^{100} ^{43} $ , and $ e^{1000} $ , which is considerably larger than the number of

atoms in the universe (if the cosmologists are to be believed). Moral: ln grows very slowly indeed.

    1. $ e^{5/2} $ b) $ e^{-2} $ c) e d) 1
    1. 0 b) 0 c) 1 d) 1 e) 4 f) -2 g) 42 h) 27 i) 0 j) 0 k) 2 l) ½
    1. $ $ b) $ $ c) $ () / $ d) $ 25 $ e) $ $
    1. [Why isn’t –2 also a solution? Try putting –2 back into the original equation, and you’ll see.]

  1. y = 10x

    1. .26 years, .33 years, .96 years b) .41 years, .59 years, 2.93 years
  1. 4.67 years, 47.21 years, $ 2.7 ^{43} $ years (!)
  1. About 0.43 years in each case. 40. Both are constant.