Science:Math Exam Resources/Courses/MATH100/December 2012/Question 03 (a)
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Question 03 (a) 

ShortAnswer Questions. Questions 14 are shortanswer questions. Put your answers in the boxes provided. Simplify your answers as much as possible, and show your work. Each question is worth 3 marks, but not all questions are of equal difficulty.

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 

First compute the equation of the tangent line. Then, how do the values of and relate to this tangent line? 
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 1 

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Please rate my easiness! It's quick and helps everyone guide their studies. We compute the equation of the line between the points and . First compute the slope via
The yintercept is given to be 5 since the point is on the line. Thus the equation of the line is . As this line is tangent to the function at 1, we see that and completing the question. 
Solution 2 

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Please rate my easiness! It's quick and helps everyone guide their studies. According to the point slope formula given by
we can use the fact that our tangent line in question contains the point and has slope and thus, we have the equation of the line is given by
Now, we know that the points and lie on our cruve and this gives the two equations
Simplifying gives
Subtracting yields
and so . Plugging back into either of the original equations yields 