9.9 Small Changes and Approximations

9.9 Small Changes and Approximations


If  δ x  is very small,  δ y δ x  will be a good approximation of  d y d x , ,

This is very useful information in determining an approximation of the change in one variable given the small change in the second variable. 


Example:
Given that y = 3x2 + 2x – 4. Use differentiation to find the small change in y when x increases from 2 to 2.02.

Solution:
y = 3 x 2 + 2 x 4 d y d x = 6 x + 2
The small change in yis denoted by δy while the small change in the second quantity that can be seen in the question is the x and is denoted by δx.

δ y δ x d y d x δ y = d y d x × δ x δ y = ( 6 x + 2 ) × ( 2.02 2 )                                                                            δ x = new  x original  x δ y = [ 6 ( 2 ) + 2 ] × 0.02                                      Substitute  x  with the original value of  x ,  i .e 2. δ y = 0.28

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