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.

δy δx dy dx δy= dy dx ×δx δy=( 6x+2 )×( 2.022 )δx=new xoriginal x δy=[ 6( 2 )+2 ]×0.02          Substitute x with the original value of x, i.e2. δy=0.28

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