Table of Contents
Can there be a tangent line at a non differentiable point?
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). Below are graphs of functions that are not differentiable at x = 0 for various reasons. (try to draw a tangent at x=0!)
When a function has a vertical tangent line at a point what must be true?
At, a, the function has a “infinite slope” or vertical tangent line. If the slope of the tangent line is considered to be the instantaneous rate of change, at that point, the function increases “straight up”.
What makes a function differentiable?
A function is formally considered differentiable if its derivative exists at each point in its domain, but what does this mean? It means that a function is differentiable everywhere its derivative is defined. So, as long as you can evaluate the derivative at every point on the curve, the function is differentiable.
Why is a function not differentiable at a cusp?
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 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. Since the slope of a vertical line is undefined, the function is not differentiable in this case.
Are functions differentiable at cusps?
Why are Functions with Cusps and Corners not Differentiable? A function is not differentiable if it has a cusp or sharp corner. As well as the problems with division by zero shown above, we can’t even find limits near the cusp or corner because the slope to the left of the cusp is different than the slope to the right.
Do tangent lines tell you if a function is differentiable?
Now, tangent lines only make sense for differentiable functions, so you shouldn’t expect that tangent lines are going to tell you whether or not something is a function. What your example would show is that such a function is not differentiable at the point where you have the vertical tangent.
When is a function not differentiable at a point?
If a function is continuous at a point, then it is not necessary that the function is differentiable at that point. A function f is not differentiable at a point x0 belonging to the domain of f if one of the following situations holds: (i) f has a vertical tangent at x 0.
Can funfunctions have a vertical tangent?
Functions DO NOT and CANNOT have VERTICAL tangents, only RELATIONS DO and CAN. Graphs of circles,ellipses,rectangular hyperbolas,parabolas, etc all have vertical tangents,AND their slope dy/dx is obtainable,and its value is infinity at points of contact of vertical tangents,and there is no problems at all with that.
What is a vertical tangent in calculus?
A vertical tangent is a line that runs straight up, parallel to the y-axis. This graph has a vertical tangent in the center of the graph at x = 0. Technically speaking, if there’s no limit to the slope of the secant line (in other words, if the limit does not exist at that point), then the derivative will not exist at that point.