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.
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.
P
Q
Q
If the secants approach the same line,
as Q approaches P from either side, this
line is called the tangent at P.
Q
Q
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
HW #1
HW #2