Science:Math Exam Resources/Courses/MATH100/December 2012/Question 10
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Question 10 

FullSolution Problems. In questions 511, justify your answers and show all your work. If a box is provided, write your final answer there. Unless otherwise indicated, simplification of numerical answers is required in these questions. Find a function such that and such that the line is tangent to the graph of . 
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 

Antidifferentiating gives you step one. However remember that antidifferentiating only gives you the answer up to a constant. We need to use the other information given to help figure our constant. How do we relate slopes and derivatives? 
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. To find the form of the function we compute the antiderivative,
To find the actual function we are interested in, we need to determine C. The extra information we are given is that we want the line , which is equivalent to to be tangent at some point on the function . The slope of this line is 1 and thus we need which happens when . Plugging this point into the line we have that and so since the line is tangent to the function there, as well. Therefore we have, Thus, our desired function is . 