298
Chapter 3 | Derivatives
255. [T] The total cost to produce x boxes of Thin Mint Girl Scout cookies is C dollars, where C =0.0001 x 3 −0.02 x 2 +3 x +300. In t weeks production is estimated to be x =1600+100 t boxes. a. Find the marginal cost C ′( x ). b. Use Leibniz’s notation for the chain rule, dC dt = dC dx · dx dt , to find the rate with respect to time t that the cost is changing. c. Use b. to determine how fast costs are increasing when t =2 weeks. Include units with the answer. 256. [T] The formula for the area of a circle is A = πr 2 , where r is the radius of the circle. Suppose a circle is expanding, meaning that both the area A and the radius r (in inches) are expanding. a. Suppose r =2− 100 ( t +7) 2 where t is time in seconds. Use the chain rule dA dt = dA dr · dr dt to find the rate at which the area is expanding. b. Use a. to find the rate at which the area is expanding at t =4 s. 257. [T] The formula for the volume of a sphere is S = 4 3 πr 3 , where r (in feet) is the radius of the sphere. Suppose a spherical snowball is melting in the sun. a. Suppose r = 1 ( t +1) 2 − 1 12 where t is time in minutes. Use the chain rule dS dt = dS dr · dr dt to find the rate at which the snowball is melting. b. Use a. to find the rate at which the volume is changing at t =1 min. 258. [T] The daily temperature in degrees Fahrenheit of Phoenix in the summer can be modeled by the function T ( x ) = 94−10cos ⎡ ⎣ π 12 ( x −2) ⎤ ⎦ , where x is hours after midnight. Find the rate at which the temperature is changing at 4 p.m. 259. [T] The depth (in feet) of water at a dock changes with the rise and fall of tides. The depth is modeled by the function D ( t ) =5sin ⎛ ⎝ π 6 t − 7 π 6 ⎞ ⎠ +8, where t is the number of hours after midnight. Find the rate at which the depth is changing at 6 a.m.
f ( x )
f ′( x )
g ( x )
g ′( x )
x
0 2
5
0
2
1 1
−2
3
0
2 4
4
1
−1
3 3
−3
2
3
245. h ( x ) = f ⎛
⎞ ⎠ ; a =0
⎝ g ( x )
246. h ( x ) = g ⎛
⎞ ⎠ ; a =0
⎝ f ( x )
−2
⎛ ⎝ x 4 + g ( x ) ⎞ ⎠
247. h ( x ) =
; a =1
⎛ ⎝ f ( x ) g ( x )
⎞ ⎠
2
248. h ( x ) =
; a =3
249. h ( x ) = f ⎛
⎝ x + f ( x ) ⎞
⎠ ; a =1
⎠ 3 ; a =2
250. h ( x ) = ⎛
⎝ 1+ g ( x ) ⎞
251. h ( x ) = g ⎛
⎛ ⎝ x 2
⎞ ⎠ ⎞ ⎠ ; a =1
⎝ 2+ f
252. h ( x ) = f ⎛ ⎠ ; a =0 253. [T] The position function of a freight train is given by s ( t ) =100( t +1) −2 , with s in meters and t in seconds. At time t =6 s, find the train’s a. velocity and b. acceleration. c. Using a. and b. is the train speeding up or slowing down? ⎝ g (sin x ) ⎞ 254. [T] A mass hanging from a vertical spring is in simple harmonic motion as given by the following position function, where t is measured in seconds and s is in inches: s ( t ) =−3cos ⎛ ⎝ πt + π 4 ⎞ ⎠ . a. Determine the position of the spring at t =1.5 s. b. Find the velocity of the spring at t =1.5 s.
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