AP Calculus BC – Chapter 3: Derivatives 3.1: Derivative of a Function Goals: Calculate slopes and derivatives using the definition of derivative.

Graph f from the graph of f’, graph f’ from the graph of f, and graph the derivative of a function given numerically.

Quote for today: “That which we must learn to do, we learn by doing.” Aristotle (382 BC – 322 BC), in Nicomachean Ethics

Definition of Derivative:  The derivative of the function f with respect to the variable x is the function f’ whose value at x is

f

   

h

lim  0

f



x



h h

provided the limit exists.

(Alternate) Definition of Derivative at a Point:  The derivative of the function f at the point x=a is the limit

f

   

x

lim 

a f x



a

provided the limit exists.

Differentiable:   The domain of f’, the set of points in the domain of f for which the limit exists, may be smaller than the domain of f. If f’(x) exists, we say that f has a derivative at x (or is differentiable at x). A function that is differentiable at every point of its domain is a differentiable function.

One-Sided Derivatives:  A function y=f(x) is differentiable on a closed interval [a, b] if it has a derivative at every interior point of the interval, and if the limits

h

lim  0 

f



a



h h

 right-hand limit

f



b



h



h

lim  0 

h

left-hand limit exist at the endpoints.

Assignment:  HW 3.1: #1, 3, 6, 9, 12-15, 17, 20, 26.

 Quiz after 3.3.