How do you find the linearization of #f(x)=x^3# at the point x=2?
1 Answer
The linearization is the equation of the tangent line. (Often presented in a different form.)
Explanation:
At
The linearization is
The equation of the tangent line
The line tangent to the graph at the point where
In point-slope form the equation is
#(y-8) = 12(x-2)#
When thinking of it as a linear approximation, we write
#y=8+12(x-2)#
More explanation
The linearization uses
The change in
On a line of slope
the change in
For the differential, we change the notation to
#dy=mdx# where#m = f(x)# at some chosen#x=a# .
#dy = f'(a)Deltax = f'(a)(x-a)#
In this question the chosen value of
so
The approximation begins at