# Calculus for Business, Economics, and the Social and Life Sciences, Brief Version (11th Edition) Chapter 2.6, Problem 44E

MANUFACTURING At a certain factory, output Q is related to inputs u and v by the equation

If the current levels of input are u = 10 and v = 25, use calculus to estimate the change in input v that should be made to offset a decrease of 0.7 unit in input u so that output will be maintained at its current level.

Solution:

At a certain factory the output Q is related to inputs u and v given as  …… (1)

We have to find the change in input v that should be made to offset a decrease of 0.7 units in input u so that output is maintained at the current level.

To find the change in input v, differentiating both the sides of (1) with respect to input u, we have  …… (2)

Since the output has to be maintained at current level,

Therefore,  From equation (2), we have  …… (3)

From the question, for inputs u = 10, v = 25, from equation (3), we have  …… (4)

Using approximation formula,  …… (5)

Therefore, from equation (4) and (5), we have

Hence decrease the input in v by 21184.30 units to maintain the output at the current level.