022 Exam 2 Sample A, Problem 8

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022 2 A 8.png

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:  
First, we must find the derivative. We have .
Step 2:  
We need the derivative at our initial point, or . This is
Step 3:  
Finally, we plug in the values to find
Note that if a teacher gives you units (thousands of dollars, dollars, cubits...), you should include them in your answer.
Final Answer:  

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