AP CALC BC formulas

Def of Critical Number

Let f be defined at c. If f'(C)=0 or is undef, then f(C) is a critical number of f.

First Derivative Test

Let c be a critical number of f on a continuous open interval of I containing c. If f is differentiable on the interval, except possibly at c, then f(C) can be classified as follows.

1) If f’(x) changes from + to -, then f(C) is a relative max of f.

2) If f’(x) changes from - to +, then f(C) is a relative min of f.

Second Derivative Test

Let f be a function such that f”(C)=0 and the second derivative of f exists on an open interval containing c.

1) If f’’(C)>0, then f(C) is a rel min.

2) If f’’(C)<0, then f(C) is a rel max.

Definition of Concavity

Let f be differentiable on an open interval I. The graph is concave upward on I is f’ is increasing on the interval and concave downward on I if f’ is decreasing on the interval.

Test for Concavity

Let f be a function whose second derivative exists on an open interval I.

1) If f’’(x)>0 for all x in I, then the graph is concave upwards in I.

2) If f’’(x)<0 for all x in I, then the graph is concave downwards in I.

Def of an Inflection Point

A function f has an inflection point at (c,f(C)))

1) If f’’(C)=0 or f””(C) does not exist.

2) If f””(C) changes sign at x=c or if f’ changes from increasing to decreasing or vice versa.