**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

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

Since the output has to be maintained at current level,

Therefore,

From equation (2), we have

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

Using approximation formula,

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.