~AVERAGE AND INSTANTANEOUS RATE OF CHANGE~

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~Generally, average rate of change, between 2 values of the independent variable, is the slope of the secant line connecting these points on the graph of the curve & instantaneous rate of change, at one value, is the slope of the tangent to the curve at that point.

~For s = f(t), this was average velocity & instantaneous velocity (or just velocity) at a pt.

~Well, these concepts can also be applied to any function, as long as we know its equation.

~Having said all that, here is a review of the procedures involved.

~Finding the average rate of change of a function between 2 values of the independent variable.

~Given a function (an equation)---this could be a formula for some geometric figure

1) use functional notation---i.e., if Volume is a function of x, use f(x) or V(x) in place of V
2) for average rate of change, we have 2 fixed values of the independent variable, say a & b, where b>a
3) compute the slope of the secant line to the function connecting the pts
(a,f(a)) and (b,f(b))

   i.e.,  f(b)-f(a)
             b-a

~Example:  A circle is expanding. Find the average rate of change
              in the area with respect to the radius as the radius  
              changes from 2 inches to 5 inches.

~Solution:  The formula used is A=pr2. Let f( r)= pr2, r=the radius

Compute  [f(5)-f(2)]/(5-2) =  [p(25)-p(4)] / 3  = 21p/3 = 7p  sqr "/ inch
                

Finding the instantaneous rate of change at a value of the independent
variable, say at r=a [the fixed pt is a,f(a)]

1) use functional notation (as in above)
2) choose a variable pt, say x,f(x)
3) formulate the slope of the secant lines as in 3) above, i.e., [f(x)-f(a)]/(x-a)

4) then take the limit as x approaches a,

 i.e.,      lim  [f(x)-f(a)]/(x-a)                                                       
           x→a

Example: In the above problem, compute the instantaneous rate of change
of area with respect to the radius when the radius is 3 inches.

Solution: compute:  lim   f(r)-f(3)  =  lim  p[(r2-9)]/(r-3)
                                r→3    (r-3)        r→3                            
    
                            lim  p(r-3)(r+3) = 6p square inches/inch
                           r→3       r-3
~Use the same exact procedure on any formula (any figure), but
make sure you have the formula in functional notation.

~Note: This is not the only approach. There are others that work as well, however, for basic problems, this one is easiest to understand, in my opinion.