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Differentiability Piecewise functions may or may not be differentiable on their domains. To be differentiable at a point

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Differentiability Piecewise functions may or may not be differentiable on their domains. To be differentiable at a point x = c , the function must be continuous, and we will then see if it is differentiable.

Let’s consider some piecewise functions first. Let f ( x ) =

− x, x ≤ 0 x,

x>0

First we will check to prove continuity at x = 0

Now we will consider differentiability at x = 0

Now let’s try: g ( x) =

8 x − 3,

x ≤1

4 x 2 + 5, x > 1

Is g (x) continuous AND differentiable at x = 1?

Is h (x) continuous and differentiable at x = 3 ? h (x) =

x 2 − 4 x + 8, x ≤ 3 2 x − 1,

x> 3

Now let’s try something trickier: If

f ( x ) = 3 x 2 + 4 x,

x ≤1

2 x 3 + bx + c, x > 1 Find b and c so that f(x) is differentiable at x = 1

Let’s work on continuity first.

Now work on differentiability

Let f ( x) =

ax 2 + 10,

x