Sec 2.2 The Derivative at a Point

Recall:

The average rate of change of any function, f, over the interval [a, a + h]

is

Instantaneous rate of change for any function, f, at some point a.

Def. The Derivative of f(x) at x = a is:

if this limit exists, is a real number, then f is differentiable at x = a.

The derivative of f(x) at x = a is notated f ’(a)

and can be interpreted as

1) The slope of the curve f(x) at the point (a, f(a))

2) The slope of the line tangent to f(x) at the point (a, f(a))

3) The instantaneous rate of change of f(x) with respect to x

at the point (a, f(a))

Ex. Sand, when piled, forms a right cone where the radius of the base equals the height of the pile.

The volume of a right cone is:

Find the rate of change of the radius with respect to the volume when the volume of the pile is 100 cubic feet.

Let’s look at the difference quotient for a small value of ‘h’: h = 0.1

Ex. Find f ’(2) where use the definition of the derivative.

Ex. Find the equation of the line tangent to the graph of at the point (1, 3)