Science:Math Exam Resources/Courses/MATH102/December 2015/Question 19 (b)
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Question 19 (b) 

In deciding how long a resident’s shift in the emergency room should be, the Chief of Staff at Vancouver General Hospital would like to minimize the average rate at which errors are made. Let be the number of errors made by a resident from the start of a shift until t hours into the shift. The instantaneous rate of change of errors made is (b) What is the total number of errors, , made hours into a shift? 
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! 
Hint 

To find , first determine . Then, use the Fundamental Theorem of Calculus to find from . 
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. It is impossible for the resident to make any errors at time , so . Next, recall that the fundamental theorem of calculus states that
where is the derivative of . Now, calculate for any as follows. Let . Then in the above equation, set , , , and . By the fundamental theorem of calculus
As , it follows that
To calculate the above definite integral, we calculate the following indefinite integral (below, recall that 48 is 16 times 3).
Now compute
Hence . By simply changing notation (write t instead of s) we can rewrite our solution as
which is the answer to the question. 