If f(3)=8 and f'(3)=-4, then how do you use linear approximation to estimate f(3.02)?

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Answer 1 · 2016-09-17T18:35:10

The linear approximation of #f# at #a# is one way of writing the equation of the tangent line at #a#.

At #x=a#, #y = f(a)# and the slope of the tangent line is #f'(a)#.

So, in point slope form, the tangent line has equation

#y-f(a) = f'(a)(x-a)#

The linearization solves for #y# by adding #f(a)# to both sides

#L = y = f(a)+f'(a)(x-a)#

In this problem, we have

#L = y = 8+(-4)(3.02-3)#

# = 8-4(0.2)#

# = 8-(0.8)#

#= 7.2#.