Quick Answer: Why Are Vertical Tangents Not Differentiable?

How do you know if a function is differentiable?

Lesson 2.6: Differentiability: A function is differentiable at a point if it has a derivative there.

Example 1: …

If f(x) is differentiable at x = a, then f(x) is also continuous at x = a.

f(x) − f(a) …

(f(x) − f(a)) = lim.

(x − a) · f(x) − f(a) x − a This is okay because x − a = 0 for limit at a.

(x − a) lim.

f(x) − f(a)More items….

What is the derivative of a vertical tangent?

If the tangent line is vertical. This is because the slope of a vertical line is undefined. 3. At any sharp points or cusps on f(x) the derivative doesn’t exist.

Is a vertical tangent a critical point?

The geometric interpretation of what is taking place at a critical point is that the tangent line is either horizontal, vertical, or does not exist at that point on the curve. … hence, the critical points of f(x) are (−2,−16), (0,0), and (2,−16).

Why are cusps not differentiable?

In the same way, we can’t find the derivative of a function at a corner or cusp in the graph, because the slope isn’t defined there, since the slope to the left of the point is different than the slope to the right of the point. Therefore, a function isn’t differentiable at a corner, either.

Why is a function continuous but not differentiable?

In particular, any differentiable function must be continuous at every point in its domain. The converse does not hold: a continuous function need not be differentiable. For example, a function with a bend, cusp, or vertical tangent may be continuous, but fails to be differentiable at the location of the anomaly.

How do you tell if a function is continuous but not differentiable?

Continuous. When a function is differentiable it is also continuous. But a function can be continuous but not differentiable. For example the absolute value function is actually continuous (though not differentiable) at x=0.

Are horizontal tangents differentiable?

Where f(x) has a horizontal tangent line, f′(x)=0. If a function is differentiable at a point, then it is continuous at that point. A function is not differentiable at a point if it is not continuous at the point, if it has a vertical tangent line at the point, or if the graph has a sharp corner or cusp.

How do you know if a tangent line is vertical?

Set the denominator of any fractions to zero. The values at these points correspond to vertical tangents. Plug the point back into the original formula. If the right-hand side differs (or is zero) from the left-hand side, then a vertical tangent is confirmed.

How do you know if a function is not differentiable?

We can say that f is not differentiable for any value of x where a tangent cannot ‘exist’ or the tangent exists but is vertical (vertical line has undefined slope, hence undefined derivative).