Flashcards Definitions for Test 2 Derivatives Calc 1

The definition of the derivative of a function at a point

The derivative of a function f, denoted by f', is the function defined by: f ' (x) = lim h->0 f(x+h) - f(x) / h - provided that the limit exists

The definition of the tangent line to a function at a point

If f ' (a) exists, then the equation of the tangent line to the graph of the function y= f(x) at the point P=(a,f(a)) is y = f ' (a) (x-a) + f(a)

An absolute maximum or absolute minimum of a function

. Absolute maximum value of f on D if f(c) >(or equal to) f(x) for all x in D

. Absolute Minimum value of f on D if f(c) < (or equal to) f(x) for all x in D

A local maximum or local minimum of a function

. Local maximum value of f if f(c) > (or equal to) f(x) for all x on some open interval containing c

. Local minimum value of f if f(c) < (or equal to) f(x) for all x on some open interval containing c

The various indeterminate forms for a limit

0/0, infinity/infinity, 0 times infinity, infinity - infinity, 0^0, infinity ^0, 1^infinity

Rolles Theorem

Let f be a function that satisfies the three hypotheses:

1. F is continuous on the closed interval [a.b]

2. F is differentiable on the open interval (a,b)

3. F(a)=F(b)

There there is a number C in (a,b) such that f'(c) = 0

The Mean Value Theorem

Let f be a function that satisfies:

1. F is continuous on [a,b]

2. F is differentiable in (a,b)

Then there exists a 'c' in (a,b), such that f'(c) = F(b)-f(a)/b-a

L' Hospitals Rule

Suppose that f and g are differentiable, g'(x) does not equal 0 near 'a' (except possibly at a), and that lim x->c has 0/0 or infinity/infinity indeterminate form, then Limx->a f(x)/g(x) = lim x->a f'(x)/g'(x)

The Extreme value theorem

If f is continuous on a closed interval [a,b], then f attains an absolute maximum value at f(c) and an absolute minimum value at f(d) at some number of c and d in [a,b]

Critical Number

a Critical number of a function f is a number 'c' in the domain of f, such that either f'(c) = 0 or f'(c) does not exist