808
Answer Key
161 . a. R ( x ) =10 x −0.001 x 2 b. R ′( x ) = 10 − 0.002 x c. $6 per item, $0 per item 163 . a. C ′( x ) =65 b. R ( x ) =143 x −0.03 x 2 , R ′( x ) = 143 − 0.06 x c. 83, −97. At a production level of 1000 cordless drills, revenue is increasing at a rate of $83 per drill; at a production level of 4000 cordless drills, revenue is decreasing at a rate of $97 per drill. d. P ( x ) =−0.03 x 2 +78 x −75000, P ′( x ) =−0.06 x +78 e. 18, −162. At a production level of 1000 cordless drills, profit is increasing at a rate of $18 per drill; at a production level of 4000 cordless drills, profit is decreasing at a rate of $162 per drill. ⎞ ⎠ ⎟ ⎟ b. 120, 0, −14.4, −9.6 c. The bacteria population increases from time 0 to 10 hours; afterwards, the bacteria population decreases. d. 0, −6, 0.384, 0.432. The rate at which the bacteria is increasing is decreasing during the first 10 hours. Afterwards, the bacteria population is decreasing at a decreasing rate. 167 . a. P ( t ) = 0.03983 + 0.4280 b. P ′( t ) = 0.03983. The population is increasing. c. P ″( t ) =0. The rate at which the population is increasing is constant. 169 . a. p ( t ) =−0.6071 x 2 +0.4357 x −0.3571 b. p ′( t ) =−1.214 x +0.4357. This is the velocity of the sensor. c. p ″( t ) =−1.214. This is the acceleration of the sensor; it is a constant acceleration downward. 171 . a. 165 . a. N ′( t ) =3000 ⎛ ⎝ ⎜ ⎜ −4 t 2 +400 ⎛ ⎝ t 2 +100 ⎞ ⎠ 2
b. f ′( x ) = a . The more increase in prey, the more growth for predators. c. f ″( x ) =0. As the amount of prey increases, the rate at which the predator population growth increases is constant. d. This equation assumes that if there is more prey, the predator is able to increase consumption linearly. This assumption is unphysical because we would expect there to be some saturation point at which there is too much prey for the predator to consume adequately. 173 . a.
⎞ ⎠ 2 . When the amount of prey increases, the predator growth increases. c. f ″( x ) = 2 an 2 ⎛
⎝ n 2 −3 x 2 ⎞ ⎠
f ′( x ) = 2 axn 2 ⎛ ⎝ n 2 + x 2
⎞ ⎠ 3 . When
b.
⎛ ⎝ n 2 + x 2
the amount of prey is extremely small, the rate at which predator growth is increasing is increasing, but when the amount of prey reaches above a certain threshold, the rate at which predator growth is increasing begins to decrease. d. At lower levels of prey, the prey is more easily able to avoid detection by the predator, so fewer prey individuals are consumed, resulting in less predator growth. 175 . dy dx =2 x −sec x tan x 177 . dy dx =2 x cot x − x 2 csc 2 x
This OpenStax book is available for free at http://cnx.org/content/col11964/1.12
Made with FlippingBook - professional solution for displaying marketing and sales documents online