022 Exam 2 Sample A, Problem 8

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Use differentials to approximate the change in profit given   units and   units, where profit is given by .

Foundations:  
A differential is a method of linearly approximating the change of a function. We use the derivative of the function at an initial point as the slope of a line, and use the standard relation
where represents the change in values, and represents the change in values. Due to the use of the derivative as the slope, we usually rewrite this using and to indicate the relative changes. Thus,
We can then rearrange this to find

 Solution:

Step 1:  
Step 2:  

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