The function given below gives the cost to manufacture x items. Estimate (using h = 0.0001)the instantaneous rate of change of the total cost at the production level .
Select your answer rounded to the nearest tenth.
A)71
B)70.2
C)70
D)71.2
E)69

Question

The function given below gives the cost to manufacture x items. Estimate (using h = 0.0001)the instantaneous rate of change of the total cost at the production level  .  
Select your answer rounded to the nearest tenth.
A)71 
B)70.2 
C)70 
D)71.2 
E)69

Quizplus · Accepted Answer

To find the instantaneous rate of change at a given point, we need to find the derivative of the total cost function at that point. Using the limit definition of a derivative, we have:

f'(11ebfe6a_2cbe_9ce3_96cc_edd9fcc92c2b_TBX8650_11) = lim h -> 0 (f(11ebfe6a_2cbe_9ce3_96cc_edd9fcc92c2b_TBX8650_11 + h) - f(11ebfe6a_2cbe_9ce3_96cc_edd9fcc92c2b_TBX8650_11)) / h

f(x) = 600 + 14x + 0.0008x^2
f(11ebfe6a_2cbe_9ce3_96cc_edd9fcc92c2b_TBX8650_11) = 600 + 14(11ebfe6a_2cbe_9ce3_96cc_edd9fcc92c2b_TBX8650_11) + 0.0008(11ebfe6a_2cbe_9ce3_96cc_edd9fcc92c2b_TBX8650_11)^2
= 600 + 154 + 133.12
= 887.12

Plugging this into the limit definition and simplifying, we get:

f'(11ebfe6a_2cbe_9ce3_96cc_edd9fcc92c2b_TBX8650_11) = lim h -> 0 (600 + 14(11ebfe6a_2cbe_9ce3_96cc_edd9fcc92c2b_TBX8650_11 + h) + 0.0008(11ebfe6a_2cbe_9ce3_96cc_edd9fcc92c2b_TBX8650_11 + h)^2 - 887.12) / h
= lim h -> 0 (14h + 0.0008h^2) / h
= lim h -> 0 (14 + 0.0008h)
= 14

Rounding to the nearest tenth, we get 14.0, which means the instantaneous rate of change of the total cost at the production level 11ebfe6a_2cbe_9ce3_96cc_edd9fcc92c2b_TBX8650_11 is approximately 14. Answer choice (C) is the closest to this value.

The Function Given Below Gives the Cost to Manufacture X