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1.5 Cusps and Corners. When we determine the derivative of a function, we are differentiating the function. For functions that are “differentiable” for all values of x, the slopes of the tangents change gradually as the point moves along the graph.

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1 5 cusps and corners
1.5 Cusps and Corners

  • When we determine the derivative of a function, we are differentiating the function.

  • For functions that are “differentiable” for all values of x, the slopes of the tangents change gradually as the point moves along the graph.

  • y=x squared is differentiable for all values of x.

  • y=x cubed is differentiable for all values of x.

  • There are some functions for which there may be points where the tangent line does not exist. The function would be not differentiable at that point.


Definition of a tangent
Definition of a Tangent

  • First we must get a better definition of a tangent.


Tangent
Tangent

  • Latin word “tangere”, which means to touch.

  • It is easy to understand this “touch” definition with the previous graphs.

  • But not all lines that “touch” a curve are tangents.


Not tangents
Not tangents

  • All these lines touch the curve at A.

  • None of them is a tangent.

  • Why?

  • Notice how abruptly the slope changes at A.

  • How do we define a tangent line?

A


Tangent defintion
Tangent Defintion

  • A tangent at a point on a curve is defined as follows:

  • Let P be a point on the curve.

P

Q

Q

  • Let Q be another point on the curve, on either side of P.

Construct the secant PQ.

Let Q get closer to P and observe the secant line.


Q on the other side

P

Q

Q

Q on the other side

  • Now let Q approach P from the other side.

  • Notice that the secant lines PQ approach the same line from both sides.

  • That is the red line and the blue line are approaching the same line.

Q

Q

If the secants approach the same line, as Q approaches P from either side, this line is called the tangent at P.


Demo of a cusp
Demo of a Cusp

  • Example of a cusp

  • Slide the green slider to change the position of point Q.

  • What is the slope of the secant as Q approaches P from the right?

  • What is the slope of the secant as Q approaches P from the left?

  • Is the function differentiable at the point P?

  • No, the function is not differentiable at point P, because the secants from either side do not approach the same line.


Derivative of the function
Derivative of the function.

  • Graph the slopes.


Example 1
Example 1

  • Graph the derivative of y = |x +2|

  • See the solution


You try
You try

  • Graph the function y = - | x –2| + 3

  • Graph the derivative.

  • See the solution:


Zooming in
Zooming In

  • If we zoom in on a function that is not differentiable the cusp or corner will always be there.

  • If we zoom in on a graph that is differentiable then the graph will eventually have a smooth curve.

  • It is a matter of the difference between P and Q being so small that we can’t even see it without zooming into the graph.

  • zoom in demo


Summary
Summary

  • What is a tangent line?

  • A function is not differentiable if it has a cusp or a corner.

  • A function is also only differentiable were it is defined.

  • So if a graph has a hole or a gap, then it not differentiable at these point.

  • There is also another situation where a function can be not differentiable – see #10 in the homework.

Step function.


Homework
Homework

  • Page 51 #1-5,8-11




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