Science:Math Exam Resources/Courses/MATH110/December 2015/Question 03 (c)
• Q1 (a) • Q1 (b) • Q1 (c) • Q1 (d) • Q2 (a) • Q2 (b) • Q2 (c) • Q2 (d) • Q3 (a) • Q3 (b) • Q3 (c) • Q4 (a) • Q4 (b) • Q4 (c) • Q5 (a) • Q5 (b) • Q5 (c) • Q5 (d) • Q5 (e) • Q6 (a) • Q6 (b) • Q6 (c) • Q6 (d) • Q7 (a) • Q7 (b) • Q7 (c) • Q8 (a) • Q8 (b) • Q8 (c) • Q9 • Q10 (a) • Q10 (b) •
Question 03 (c) 

Consider the following function,
where and are constants. (c) At which point(s) is the function drawn above differentiable? Justify your answer. 
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 

Check differentiability for the points where the pieces meet each other, i.e. at and . 
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 

Found a typo? Is this solution unclear? Let us know here.
Please rate my easiness! It's quick and helps everyone guide their studies. We recall that a function is NOT differentiable at a point if the limit doesn't exist. Also, if a function has a corner, a cusp or a vertical tangent line for the graph, at that point the function is NOT differentiable.
To check this by using the definition, we first calculate the left hand limit . The first equality follows from that we consider negative and for such , . However, we have the right hand limit . Since the left hand limit doesn't match with the right hand one, the limit doesn't exist.
and the right hand limit . Therefore, the function is differentiable at .
