Sage-Tips

Are all absolute value functions non differentiable?

The left limit does not equal the right limit, and therefore the limit of the difference quotient of f(x) = |x| at x = 0 does not exist. Thus the absolute value function is not differentiable at x = 0. So, for example, take the absolute value function f(x) = |x| and restrict it to the closed interval [−1, 2].

Why Sharp edges are not differentiable?

Since , this limit does not exist, and so the derivative of the absolute value function at does not exist. In fact, this is a so-called corner point. On the graph of this function, there is a sharp corner at . This signifies that the slope of the function on one side of is different on the other side.

How do you know if a function has a sharp corner?

Sharp corner: The graph is continuous, but slopes on either side of the sharp corner do not approach each other. A continuous piecewise function (e.g., an absolute value function) may have this behavior.

Is a sharp corner differentiable?

A function can be continuous at a point, but not be differentiable there. In particular, a function f is not differentiable at x=a if the graph has a sharp corner (or cusp) at the point (a, f (a)).

Where is a function not differentiable?

A function is not differentiable at a if its graph has a vertical tangent line at a. The tangent line to the curve becomes steeper as x approaches a until it becomes a vertical line.

Where is absolute value differentiable?

The absolute value function is continuous (i.e. it has no gaps). It is differentiable everywhere except at the point x = 0, where it makes a sharp turn as it crosses the y-axis. A cusp on the graph of a continuous function.

Why is there no derivative at a sharp point?

A geometric answer: At a sharp corner, there are many possible tangent lines; any line that (locally) intersects the curve only at the corner point meets the geometric definition of a tangent. These lines will have slopes in the closed interval between the two one-sided limits approaching the corner point.

Why a function is not differentiable at corner point?

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.

What is a sharp corner?

Adj. 1. sharp-cornered – having sharp corners. sharp-angled. angulate, angular – having angles or an angular shape.

How do you tell if a graph has a corner?

A corner is one type of shape to a graph that has a different slope on either side. It is similar to a cusp. Here, the derivative at x=0 is undefined, because the slope on the left side is 1 , but the slope on the right side is −1 .

Which all functions are not differentiable?

Generally the most common forms of non-differentiable behavior involve a function going to infinity at x, or having a jump or cusp at x. There are however stranger things. The function sin(1/x), for example is singular at x = 0 even though it always lies between -1 and 1.

How to graph the absolute value of a function?

Recall that the absolute value of a number is its distance from 0 on the number line. The absolute value parent function, written as f ( x) = | x |, is defined as. To graph an absolute value function, choose several values of x and find some ordered pairs. Plot the points on a coordinate plane and connect them. Observe that the graph is V-shaped.

When is a function differentiable at x 0?

The function is differentiable from the left and right. As in the case of the existence of limits of a function at x 0, it follows that if and only if f’ (x 0 -) = f’ (x 0 +). If any one of the condition fails then f’ (x) is not differentiable at x 0.

Why is the absolute value of a number not differentiable?

It is so because absolute value is not in a variable form and the differentiation of any constant number without any variable is always equal to zero. So this value is a numerical form that is non differentiable.

How do you find the absolute value of a number?

Recall that the absolute value of a number is its distance from 0 on the number line. The absolute value parent function, written as f ( x) = | x |, is defined as