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1.5 Cusps and Corners

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- 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.

- First we must get a better definition of a 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.

- 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

- 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

- 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.

- 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.

- Graph the slopes.

- Graph the derivative of y = |x +2|
- See the solution

- Graph the function y = - | x –2| + 3
- Graph the derivative.
- See the solution:

- 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

- 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.

- Page 51 #1-5,8-11