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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.
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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