Changes

 

 

 

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Rates of Change

 

Maximization

  

Small Changes

 

Quiz

  

Calculus Main Page

 

 

Small Changes

 

We already know the definition of the derivative,
When dx is small,
                           
This is only an approximation and only works for small dx.
 
Given y = x3, find the approximate change in y when x increases from 9 to 9.01.
 
Solution:
 
                dy/dx = 3x2
                    dx = 9.01 - 9 = 0.01
when x = 9,
                dy/dx = 243
 
                               dy = 243 x 0.01 = 2.43
 
The idea of small changes can be represented graphically.
 
dy is represented by the blue line, the estimated value. The actual value is represented by the red line. Thus it can be seen that the value we obtained in the above calculations is an underestimate.
 
The above diagram is a magnification of the actual graph. However, it can be understood that the change in x, dx, must be small, otherwise the estimated value of dy will be too far from the actual value.