3.2 Differentiability

Photo by Vickie Kelly, 2003 Greg Kelly, Hanford High School, Richland, Washington 3.2 Differentiability Arches National Park

To be differentiable, a function must be continuous and smooth. Is the function continuous? = 2 Answer: Yes = 2

But by definition, what is a derivative? A LIMIT or This is another reason why we need to keep the definition of the derivative in mind before moving on to the short cuts for derivatives.

Differentiable also means that the left and right limits of the derivative are equal. Is the function Differentiable at x = 1? = 3 But a limit only exists if it’s left and right limits are equal.

Differentiable also means that the left and right limits of the derivative are equal. Is the function Differentiable at x = 1? = 2 Answer: No ≠

 To be differentiable, a function must be continuous and smooth. Is the function Differentiable at x = 1? Notice that on the left side, the slope is approaching 3 ≠ While on the right side, the slope is 2

To be differentiable, a function must be continuous and smooth. Is the function differentiable at x = 1? First of all, is the function continuous? Yes Now can we show if it is differentiable at x = 1?

To be differentiable, a function must be continuous and smooth. Is the function differentiable at x = 1? =2 =2

To be differentiable, a function must be continuous and smooth. Is the function differentiable at x = 1? Notice that on the left side, the slope is approaching 2 Therefore, the answer is YES. While on the right side, the slope is 2

To be differentiable, a function must be continuous and smooth. Derivatives will fail to exist at: corner cusp discontinuity vertical tangent

Most of the functions we study in calculus will be differentiable.

If f has a derivative at x = a, then f is continuous at x = a. There are two theorems on page 110: Since a function must be continuous to have a derivative, if it has a derivative then it is continuous.

If a and b are any two points in an interval on which f is differentiable, then takes on every value between and . Between a and b, must take on every value between and . Intermediate Value Theorem for Derivatives p

3.2 Differentiability