Last update: 13 August 2014
A function is continuous at if it doesn't jump at i.e. if Not continuous at Think about in terms of the graph slope of at the point
A function differentiable at if the derivative exists, i.e. if the slope of the graph of at exists.
Graph Then So is not differentiable at
Graph Notes:
(a) | is the same as |
A function is increasing at
if it is going up at
i.e. if
for all small
i.e. if slope is positive,
i.e. if
A function is decreasing at
if it is going down at
i.e. if for all small
i.e. if the slope of at is negative,
i.e. if
is concave up at if it is
right side up bowl shaped
i.e. if the slope of is getting larger at
i.e. if is increasing at
i.e. if
is concave down at if it is
upside down bowl shaped
i.e. if the slope of is getting smaller,
i.e. if is decreasing at
i.e. if
A point of inflection is a point where changes from concave up to concave down, or from concave down to concave up.
A local maximum is a point where is bigger then the around it.
A local minimum is a point where is smaller then the around it. i.e. for small
A critical point is a point where a maximum or minimum might occur.
Note:
(1) | If is continuous and differentiable and is a maximum then |
(2) | If is continuous at is differentiable at then is a minimum. |
Where can a maximum or minimum occur?
(a) | A point where is differentiable and |
(b) | A point where is not continous. is a maximum. |
(c) | A point on the boundary of where is defined. is a minimum. |
These are a typed copy of Lecture 15 from a series of handwritten lecture notes for the class MATH 221 given on October 11, 2000.