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INSTANTANEOUS RATE OF CHANGE AS A LIMIT OF AVERAGE RATE OF CHANGE

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Introduction

instantaneousrate
In this topic we will discuss about real-world phenomena involving change in quantities – speed of a rocket, inflation of currency, number of bacteria in a culture etc.  Here we will interpret both average and instantaneous velocity geometrically and we will define the slope of curve at a point.  In this section we will explore the connection between velocity at an instant, the slope of a curve at a point and rate of change.

If an object moves along an s-axis and if the position versus time curve is s=f(t), then the average velocity of the object between times t0 and t1

                               instantaneousrate1
It is represented geometrically by the slope of the line joining the points   (t0, f(t0)) and (t1, f(t1)).

Slope of a secant line

                              instantaneousrate2
Consider the function y=f(x) whose graph is shown in the figure.  The line through two points on the curve is called a secant line.  Here PQ is a secant line.  We know to calculate the slope of a line through two points.  Let P(x0, f(x0)) and Q(x1, f(x1)) be the given points then slope of the secant line PQ is given by

                        instantaneousrate3
As the sampling point Q(x1, f(x1)) is chosen closer to P, that is, as x1 is selected closer to x0, the slopes msec more nearly approximate what might reasonably call the slope of the curve y = f(x) at the point P. Thus from the above equation slope of the curve y=f(x) at P(x0, f(x0)) is defined by

                       instantaneousrate4
Example: Consider the function f(x) = 6x-x2 and the point P(2, f(2)) = (2, 8).
a) Find the slopes of secant lines to the graph of y=f(x) determined by P and points on the graph at x=3 and x=1.5
b) Find the slope of the graph of y=f(x) at the point P.
Solution:

a) The secant line to the graph of ‘f’ through P and Q(3, f(3)) = (3, 9) has slope
                              instantaneousrate5
The secant line to the graph of ‘f’ through P and Q(1.5, f(1.5)) =(1.5, 6.75) has the slope
                               instantaneousrate6
b) The slope of the graph of f at the point P is

                             instantaneousrate7


Geometrical interpretation of instantaneous      velocity

If a particle moves along an s-axis, and if the position versus time curve is   s =f(t), then the instantaneous velocity of the particle at time t0,

                                   instantaneousrate8

It is represented geometrically by the slope of the curve at the point        (t0, f(t0)).
In general, if x and y are quantities related by an equation y=f(x), we can consider the rate at which y changes with x.  As with velocity, we distinguish between an average rate of change, represented by the slope of secant line to the graph of y=f(x), and an instantaneous rate of change, represented by the slope of the curve at a point.

Slopes and rates of change

If y=f(x), then the average rate of change of ‘y’ with respect to ‘x’ over the interval [x0, x1] is
                                instantaneousrate9
Geometrically, the average rate of change of y with respect to x over the interval [x0, x1] is the slope of the secant line to the graph of y=f(x) through the points (x0, f(x0)) and (x1, f(x1))                                        
                               instantaneousrate10

If y=f(x), then the instantaneous rate of change of ‘y’ with respect to ‘x’ when x=x0 is

                             instantaneousrate11
Geometrically, the instantaneous rate of change of y with respect to x when x=x0 is the slope of the graph of y=f(x) at the point (x0, f(x0)).
                            
                           instantaneousrate12


  1. Example: Let y=x2+1
    a)    Find the average rate of change of y with respect to x over the interval [3, 5]
    b)    Find the instantaneous rate of change of y with respect to x when x=-4
    Solution: a) Given f(x) = x2+1, x0=3 and x1=5

                           instantaneousrate13
    Thus, on an average, y increases 8 units per unit increase in x over the interval [3, 5].
    b)                  instantaneousrate14
    Thus, for a small change in x from x=-4, the value of y will change approximately eight times as much in the opposite direction, since the instantaneous is negative, the value of y decreases as values of x move through x=-4 from left to right.

    Rates of change in Applications

    In applied problems, average rate of change must be accompanied by appropriate units.  In general, the units for a rate of change of y with respect to x are obtained by “dividing” the units of y by the units of x and then simplifying according to the standard rules of algebra.  Here are some examples:
    a)    If y is in degrees Fahrenheit (⁰F) and x is in inches )in), then a rate of change of y with respect to x has units of degrees Fahrenheit per inch (⁰F/in)
    b)    If y is in feet per second (ft/s) and x is in seconds(s), then a rate of change of y with respect to x has units of feet per second per second (ft/s/s), which would usually be written as ft/s2

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