linear approximation

Question

Use linear approximation, i.e. the tangent line, to approximate 2.7³ as follow:

Let f(x)=x³ . The equation of the tangent line to f(x) at x=3 can be written in the form y=mx+b

 

where m is = 

 

and where be is =

 

find approximation for 2.7³ is =

Answer 1

Look for a whole number that is close to 2.7.  

 

x₀ = 3

 

 

Then you want to use the linear approximation formula

 

L(x) = f(x₀) + f'(x₀)(x - x_o)

 

f(x₀) = 3³ = 27

 

f'(x_o) = 3x² = 3(3)² = 27

 

 

Plugging in these values,

 

L(x) = 27 + 27(x - 3)

L(x) = 27(1 + (x - 3))

L(x) = 27(x - 2)

L(x) = 27x - 54

 

Evaluate L(2.7).