The Derivative and Rates of Change
MAT 135: Calculus
THE STORY SO FAR: 
Rate at which a function is changing at x=a
= Instantaneous rate of change at x=a
= Instantaneous velocity at time x=a
Want to measure the rate at which a function changes
> Can be different at different points. 
To measure instantaneous R.O.C. at x=a: 
Slope of tangent line to graph of function at x=a
To CALCULATE SLOPE OF TANGENT LINE
TO GRAPH OF FUNCTION AT X=A:
Visual estimation using graph
Points on tangent line: (1, 130) and (3,50) (?)
Slope is (130-50)/(1-3) = -40
Numerical estimation using table
As second point on secant line approaches 1, slopes of secants approach -40. 
So slope of tangent line = -40 (?). 
Algebraic calculation using limit laws
1: Calculate f(a)  (where a is point of tangency)
2: Choose variable second point x; calculate f(x)
3: Calculate slope of secant: (f(x)-f(a)) / (x-a)
4: Move x closer to a and recalculate (3)
5: Continue until limit becomes apparent. 
The DERIVATIVE of a function f(x)
at a point x= a....