Science:Math Exam Resources/Courses/MATH100/December 2011/Question 01 (n)
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Question 01 (n) 

ShortAnswer Questions. Put your answer in the box provided but show your work also. Each question is worth 3 marks, but not all questions are of equal difficulty. Full marks will be given for correct answers placed in the box, but at most 1 mark will be given for incorrect answers. Unless otherwise stated, it is not necessary to simplify your answers in this question. Find a function f(x) and a number a such that 
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 

What is the limit definition of f'(x)? Where is it similar to the above expression? 
Hint 2 

It may be useful to know that or equivalently 
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. The limit definition of the derivative at a point x=a is: We notice that the term cos(/3 + h) in the statement of the question resembles the f(x+h) term in the limit definition of the derivative. If this is the case, f(x) = cos(x) and a = /3. We now test these guesses using the limit definition of the derivative. If f(x) = cos(x) and a = /3, then If we multiply both sides by 2 we get This is the expression we were trying to find, so our guess of the function f(x) and the value a were correct: and . 