Chapter 4.03: How Derivatives Affect the Shape of a Graph

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QA: Sketch a function and give an example formula for a function that is:
i) concave up, increasing
ii) concave down, decreasing
iii) concave up, decreasing
iv) concave down, increasing
This is a good set of things to have memorized.

QB: The book defines concave up as "if the graph of f lies above all of its
tangents". Prof. Ross prefers the following definition:

f is concave up if every secant line segment falls on or above the graph.

That is, for any two points (a,f(a)) and (b,f(b)) the line segment that
connects the two points stays on or above f(x) between x=a and x=b.
Draw a function that is concave-up using this better definition that
is _not_ concave-up using the book's definition. Hint: piecewise!
Explain why it is CU using the better definition but not the book's
definition.

QC: Consider a linear function f(x)=m*x+b.
i) Is it CU according to the book's definition?
ii) Is it CU according to the better definition given in QB?
iii) Is it CD according to the book's definition?
iv) Is it CD according to the better definition given in QB
(replace "above" with "below" to get the better definition of CD).
v) Would you say that a line is both CU and CD, or neither CU nor CD?
This is an important issue in multivariable optimization theory
(Math 319 and Math 560).