3.1 Derivatives

Chapter 3 begins differential calculus which deals with rates of change.  This enables finding the slope of tangent lines and the velocity of moving objects, among many other things

Definition 1

The tangent line to the curve y = f(x) at the point P(a, f(a)) is the line through P with slope  provide that this limit exists.

Example

Write the equation of the tangent line at x = 1.    Use a = 1

Definition 4

The derivative of a function f(x) at a number, a, denoted f'(a) is

if the limit exists.

Alternatively

Example       Find the derivative of the function  at x = a using both forms of the definition.

The derivative f(a) is the Instantaneous Rate of change of y = f(x) with respect to x when x is a.

Consider visual interpretation if f '(a) is

Positive                         versus                  Negative

Large                                      versus                  Small

Zero

Example    #2   Estimate graphically the slope of the tangent line  (TRANSPARENCY)

Velocity  If s = f(t) is a position function,  f '(a) is the velocity at  t = a.

Speed    | f '(a)|

Example

A particle moves along a straight line with equation of motion , where s is measured in meters and t in seconds.  Find the velocity when t = 2.

If, find  f '(1) and find and graph the equation of the tangent line.

If, find f ‘(a)

If, find f '(a)

Each limit represents the derivative of some function f at some number a.  State such an f and a in each case.  Identify the function from the definition of f'(a)

#20                                                #22

#24

Finding derivatives with the limit definition is very tedious.  Soon patterns will emerge and some simpler techniques will become available to find derivatives, but students are expected to be able to find derivatives using the definition.

Assignment:  pg. 119; 1,3, 9a,b, 13, 17, 21, 25, 27, 31-37 odd