Concepts For Shaping A Graph

Front 1

Increasing/Decreasing Test

Back 1

(a) If f '(x) > 0 on an interval, then f is increasing on that interval.

(b) If f '(x) < 0 on an interval, then f is decreasing on that interval.

Front 2

The First Derivative Test
(determining local extrema)

Back 2

Suppose that c is a critical number of a continuous function f.

(a) If f ' changes from positive to negative at c, then f has a local maximum at c.

(b) If f ' changes from negative to positive at c, then f has a local minimum at c.

(c) If f ' does not change sign at c (for example, if f ' is positive on both sides of c or negative on both sides), then f has no local maximum or minimum at c.

Front 3

Concavity

Back 3

If the graph of f lies above all of its tangents on an interval I, then it is called concave upward. If the graph of f lies below all of its tangents on I, it is called concave downward on I.

Front 4

Concavity Test

Back 4

(a) If f "(x) > 0 for all x in I, then the graph of f is concave upward on I.

(b) If f "(x) < 0 for all x in I, then the graph of f is concave downward on I.

Front 5

The Second Derivative Test
(determining local extrema)

Back 5

Suppose f " is continuous near c.

(a) If f '(c) = 0 and f "(c) > 0, then f has a local minimum at c.

(b) If f '(c) = 0 and f "(c) < 0, then f has a local maximum at c.

Front 6

Inflection Point

Back 6

A point P on a curve y = f (x) is called an inflection point if f is continuous there and the curve changes from concave upward to concave downward or from concave downward to concave upward at P.