I. Definitions:

A function is increasing on an interval I if x₁ <x₂ => F(x₁)<F(x₂) for any x_1, x₂ in I.

A function is decreasing on an interval I if ______________________for any x_1, x₂ in I.

A function is concave up on an interval I if, over that interval, the graph:

· “curves upwards”

· a line segment connecting any two points lies above the graph

· any tangent lies below the graph

A function is concave down on an interval I if:

·

·

·

An inflection point is a point on the graph of a function where the function is continuous and

changes concavity.

II. What do derivatives tell you about the shape of the graph of a function F(x)?

1. First derivative (=slopes of the graph) determines:

1. The sign of the first derivative tells if the function is increasing or decreasing:

i. F’(c)>0 means that F is ____________________at c.

ii. F’(c)<0 means that F is ____________________ at c.

2. F’(c)=0 (or F’(c) undefined) means that c is a critical point for F

2. Second Derivative (=how the slopes are changing) determines:

1. The sign of the second derivative tells the concavity of the graph

i. F’’(c)>0 means that the function is concave up J (because the slopes are increasing)

ii. F’’(c)<0 means that the function is concave down L (because _______________________________)

2. F’’(c)=0 means that either c is a critical point, or it’s an inflection point.