Science:Math Exam Resources/Courses/MATH110/December 2010/Question 07
• Q1 (a) • Q1 (b) • Q1 (c) • Q2 (a) • Q2 (b) • Q2 (c) • Q3 (a) • Q3 (b) • Q3 (c) • Q4 (a) • Q4 (b) • Q5 (a) • Q5 (b) • Q5 (c) • Q6 (a) • Q6 (b) • Q6 (c) • Q7 • Q8 (a) • Q8 (b) • Q9 (a) • Q9 (b) • Q9 (c) • Q9 (d) • Q9 (e) • Q9 (f) • Q10 •
Question 07 

A function is defined piecewise by Find values of a and b so that is continuous and differentiable everywhere. 
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 

A function is continuous at a point x = a if its limits from both sides of a exist and the value of these limits agrees with . Furthermore, the function is differentiable at a if the derivatives from both sides of a exist and agree at a. 
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. It is clear that this function is continuous and differentiable everywhere, except maybe at x = 1. Let's start with checking continuity. For f(x) to be continuous at x = 1 the left and right limits of the function must be the same. Hence we set Because these functions are polynomials we can simply substitute the xvalue to get: This gives us one equation in terms of a and b. We can't proceed any further with this equation right now, so we will turn to the second condition that the function be differentiable. This means that the derivatives from the right and left of x = 1 must be equal. In other words:
Plugging in x = 1 this yields which gives another equation in terms of a and b. We know both equations must hold if is to be continuous and differentiable, so we can solve them as a system of equations to find a and b. Solving the first equation for b we get: Substituting this into the second equation we get: Plugging this back into we get that . (Note that the system of equations can be solved multiple ways  using a different substitution or elimination would also work.) So and are values that satisfy the conditions that be continuous and differentiable. 