# 1.5 Cusps and Corners - PowerPoint PPT Presentation

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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

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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.

• 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

• First we must get a better definition of a 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

• 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

• 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.

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

• 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.

• Graph the slopes.

### Example 1

• Graph the derivative of y = |x +2|

• See the solution

### You try

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

• Graph the derivative.

• See the solution:

### 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

• 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

• Page 51 #1-5,8-11