1.5 Cusps and Corners

1. 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.

2. Definition of a Tangent • First we must get a better definition of a tangent.

3. 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.

4. 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

5. 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.

6. 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.

7. 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.

8. Derivative of the function. • Graph the slopes.

9. Example 1 • Graph the derivative of y = |x +2| • See the solution

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

11. 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

12. 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.

13. Homework • Page 51 #1-5,8-11

14. HW #1

15. HW #2