MATH110 December 2012
• Q1 (a) • Q1 (b) • Q2 (a) • Q2 (b) • Q2 (c) • Q2 (d) • Q2 (e) • Q2 (f) • Q3 (a) • Q3 (b) • Q3 (c) • Q3 (d) • Q4 • Q5 • Q6 • Q7 (a) • Q7 (b) • Q8 • Q9 (a) • Q9 (b) • Q9 (c) • Q10 •
Find constants and such that the function is differentiable on the interval . Justify your answer.
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In order for a function to be differentiable over the domain, it should be differentiable within each piece of the function as well as on the boundary. What are the conditions for a function to be differentiable at a point? How is differentiability related to continuity?
In order for a function to be differentiable at a point, the derivative must be equal on both sides of the point. Also, differentiable functions are also continuous, meaning that the value of the function must also be equal on both sides of any point.
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In order for to be differentiable, the derivative at any point must be equal from both the left and right. Since is clearly differentiable at all points in other than x = 1, our point of interest is .
The derivative of at x = 1 from the left will be
which when evaluated at x = 1 gives
The derivative of at x = 1 from the right will be
To be differentiable, these must be equal, so
This gives us the value for a. Now we also know that in order for the function to be differentiable, it must also be continuous. In particular, this means both the right and left hand limits at x = 1 must be equal. So we have:
So will be differentiable when a = 1 and b = -1/2
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