Science:Math Exam Resources/Courses/MATH100 A/December 2023/Question 16
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Question 16 

Use the linear approximation of at to estimate . 
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 hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint. 
Hint 1 

Recall that the linear approximation to a function at a point is . 
Hint 2 

Use the expression for linear approximation given in the previous hint, how can you choose the point such that is easy to evaluate? 
Checking a solution serves two purposes: helping you if, after having used all the hints, 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. Recall from the hint the linear approximation of a function near a point is . Our function is called , so that’s what we’ll use now. We want to estimate , so, in linear approximation terms, we want to calculate where . To do this, we need to pick an appropriate so that it’s close enough to , but also so that we can evaluate in our heads the number . Since is very close to , and we know , let’s pick . Now we evaluate: , and this is our approximation. . 