linear approximation

Question

The linear approximation at x=0 to √(6+5x) is A+Bx where A is:?? and B is:??

Answer 1

Recall the linear approximation of f(x) in the neighborhood of x=a is

 

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

 

In this case, f(x) = √(6+5x) = (6+5x)^1/2 so:

 

f'(x) = (6+5x)^-1/2 (6+5x)' / 2 = 5/[2√(6+5x)]

 

So f(0) = √6 and f'(0) = 5/(2√6).

 

We have:

 

L(x) = f(0) + f'(0)(x - 0) = √6 + [5/(2√6)]x

 

A = √6 ; B = 5/(2√6)